Geodesic dominated coloring in certain product graphs

Abstract

A geodesic dominated coloring of a graph G is a proper coloring in which each color class is dominated by at least one geodesic. The minimum number of colors required for such a coloring is the geodesic dominated chromatic number, denoted by . In this paper, we determine the geodesic dominated chromatic number for several families of graphs constructed through standard product operations, including the lexicographic product, the vertex corona, and the Hamiltonian edge corona. We further establish general bounds and characterize the influence of structural properties of these product graphs on .

1 Introduction

Graph coloring is a standard and widely studied topic in graph theory with numerous theoretical and practical applications. For a positive integer k, let [k] = {1, 2, …, k}. A function c:V(G) → [k] is a proper coloring of G provided that no edge uv∈E(G) has c(u) = c(v). The minimum integer k for which G admits a proper vertex coloring is called the chromatic number of G, denoted by χ(G). When the meaning is unambiguous, the term proper coloring will be used in place of proper vertex coloring. For a fixed color i, the set of vertices assigned that color is called the color class of i, and is written as C(i) = {v∈V(G)∣c(v) = i}. Since adjacent vertices are required to receive distinct colors, every such set C(i) contains only mutually non-adjacent vertices. Hence, each color class is an independent set, and the family {C(1), C(2), …, C(k)} forms a partition of V(G). The greatest number of vertices that can occur in any independent set of G is called the independence number, denoted by α(G).

Graph coloring is a fundamental technique used to model and solve various combinatorial optimization problems. Its theoretical studies in this field are deeply rooted in the exploration of the chromatic number and the development of efficient coloring algorithms. In research, there are numerous methods of graph coloring methods and techniques. In Matula et al. [1], Matula et al. provide significant work on algorithms related to graph coloring. Some of the techniques are Approximation Algorithms [2], Parameterized Complexity [3], Probabilistic Methods [4], Hybrid Algorithms [5–7], Structural Decomposition [8]. The methods and techniques demonstrate that the strategies depend on how graph coloring is being tackled. There is insightful knowledge, allowing us to gain understanding and solve the fundamental problem of graph theory. In contrast to graph coloring, domination represents a fundamental concept in graph theory that concerns the control of vertices by specific subsets. A subset S⊆V(G) is a dominating set of G whenever every vertex outside S has a neighbor in S. The parameter γ(G) is the least cardinality of such a set. The foundational concepts were initially established by Ore [9] and Berge [10], whose pioneering works significantly influenced subsequent research in this field. A dominating set in graph theory, as it can be used to solve various problems entirely around network design for achieving communication coverage and resource allocation. It has wide applications in relay node placement, scheduling, and image processing. It also helps understand the building blocks of a graph, which contributes to theoretical and problem-solving skills. Significant related developments concerning this problem have been presented in Rao et al. [11], Garg et al. [12], Meybodi et al. [13], Manuel and Parthiban [14], Jena and Wojciechowski [15], and Kawano [16].

Beyond the classical coloring and domination problems, researchers have explored their interplay, giving rise to the intriguing concept of dominated coloring. This approach unifies coloring and domination, opening new avenues for theory and applications. A dominated coloring of G is a proper coloring in which each color class admits a vertex adjacent to all vertices in that class. The parameter χ_dom(G) is the least cardinality of such a set. The concept of dominated coloring was first introduced by Merouane et al. [17] in 2015. In Chen [18], Chen et al. discuss the application of dominated coloring in network optimization, highlighting its utility in managing interpersonal relationships within social networks and other networked systems. Significant related developments concerning this problem have been presented in Choopani et al. [19], Goddard and Henning [20], Li et al. [21], Poonkuzhali and Jayagopal [22], and Shalu et al. [23, 24].

A geodesic in a graph G is a shortest path connecting two vertices. For a geodesic λ, its closure is defined as I(λ) = ⋃_x∈V(λ)N[x], that is, the union of the closed neighborhoods of all vertices lying on λ. We say that a geodesic λ dominates a vertex set S⊆V(G) if S⊆I(λ). A supreme geodesic is a geodesic whose closure contains the maximum number of vertices among all geodesics in G. A geodesic dominated coloring of a graph G is a proper coloring in which, for every color class, there exists a geodesic that dominates it. In other words, for every color class C of G, there exists a geodesic λ in G such that C⊆I(λ). The geodesic dominated chromatic number represents the minimum number of colors needed for a geodesic dominated coloring of G. Throughout the paper, GDC refers to geodesic dominated coloring. In Vathana MP, Jayagopal [25], the concept of GDC was introduced, and the geodesic dominated chromatic number was determined for various network structures such as cycles, hypermeshes, complete binary trees, sibling trees, and hypertrees. Moreover, several lower bounds on the geodesic dominated chromatic number were established.

We organize the rest of the paper as follows:

• Section 2: We discuss the lexicographic product, the vertex corona, and the Hamiltonian edge corona, outlining their definitions and structural relevance as a basis for analyzing GDC.

• Section 3: We briefly present the conceptual background and motivation of GDC, highlighting its interplay with domination theory, geodesic concepts, and proper vertex coloring, and explaining why shortest paths naturally regulate color classes in network-based applications.

• Section 4: We present some known results that establish relationships between geodesic dominated chromatic number and other graph parameters, including bounds based on the chromatic number, domination, and properties of supreme geodesics.

• Section 5: We determine the exact geodesic dominated chromatic numbers for several graph classes formed via standard graph products, including the lexicographic, vertex corona, and Hamiltonian edge corona, and establish general structural bounds for these products.

• Section 6: We summarize the key findings on domination-based coloring via geodesics in product graphs, highlighting exact values, general bounds, and the role of structural properties in determining the geodesic dominated chromatic number. We also outline potential directions for future research on the studied problem.

2 Basic preliminaries

Product graphs provide a unifying framework in graph theory for modeling complex systems by combining simpler structures through well-defined operations. In this work, we focus on the lexicographic product, the vertex corona, and the Hamiltonian edge corona (introduced here), each offering distinct structural features for studying geodesic properties and coloring strategies. The lexicographic productG•H of graphs G and H is the graph whose vertices are of the form (u, x), where u∈V(G) and x∈V(H). Two vertices (u, x) and (v, y) are adjacent in G•H if and only if either uv∈E(G), or u = v and xy∈E(H). The lexicographic product of graphs was first introduced by Harary [26], where it was studied from a group-theoretic perspective in the context of automorphism groups. A more systematic combinatorial treatment of the lexicographic product, including its structural properties and algebraic behavior, was later developed by Sabidussi [27]. This product is ideal for modeling decision-making processes where decisions and options are layered, such as in scheduling, prioritization in task networks, or complex decision trees [28]. The vertex coronaG°H is the graph formed from one copy of G and |V(G)| mutually disjoint copies of H, where each vertex of G is made adjacent to all vertices of exactly one associated copy of H. For a vertex v∈V(G), we denote by H_v the copy of H attached to v in G°H. In Frucht and Harary [29], Frucht and Harary introduce a new graph operation called the corona, where the resulting graph's automorphism group is generally isomorphic to the wreath product of the automorphism groups of the original graphs. Let G be a Hamiltonian graph and let 𝒞 be a fixed Hamiltonian cycle of G. The Hamiltonian edge coronaG♦H is formed by taking one copy of G together with |E(𝒞)| pairwise disjoint copies of H, indexed by the edges of 𝒞. For each edge (u, v)∈E(𝒞), we attach the corresponding copy of H by adding all edges from each of u and v to every vertex in that copy. We introduce this construction as a new graph operation designed to investigate structural and coloring properties arising from Hamiltonicity. For two vertex-disjoint graphs, the join is formed by combining the two graphs and then adding every edge that connects a vertex of one graph to a vertex of the other. The union of two graphs is the graph obtained by collecting all the vertices and all the edges from both graphs. The intersection of two graphs is the graph determined by the vertices and edges common to both.

3 Conceptual background and motivation

Geodesic dominated coloring (GDC) lies at the intersection of three classical themes in graph theory: domination, geodetic structure, and proper vertex coloring. Domination captures the idea of control or coverage through a strategically chosen set of vertices, while geodetic concepts incorporate the geometry of shortest paths and reflect how information, influence, or monitoring propagates efficiently through a network. Formally, a set S⊆V(G) is geodetic whenever every vertex outside S lies on some shortest path between a pair of vertices from S, and it is strong geodetic if this containment occurs on a unique shortest path; the minimum such sizes are g(G) and sg(G), respectively. When the same set also dominates G, one obtains the corresponding hybrid parameters: a geodetic dominating set [of minimum size γ_g(G)] and a strong geodetic dominating set [of minimum size γ_sg(G)]. These notions are fundamentally vertex-centric: one seeks a small collection of vertices that, through adjacency and shortest-path structure, govern the entire graph.

GDC departs from this vertex-centric viewpoint by imposing domination requirements on color classes. In GDC, the vertex set is first partitioned into independent sets (a proper coloring), and then each color class must be “controlled” through shortest-path structure: for every color class, there exists a geodesic whose geodetic closure contains that entire class. Thus, shortest paths serve as structured “controllers” that dominate groups of vertices rather than individual vertices. This class-based formulation is important in contexts where vertices are naturally grouped into roles, tasks, frequencies, time slots, or operational clusters, and where supervision is performed along efficient routes. Accordingly, the optimization objective changes: instead of minimizing the size of a dominating-type vertex set, one minimizes the number of colors subject to a distance-aware domination constraint.

From an applied perspective, many network-design requirements can be viewed as a single composite problem with multiple constraints. First, the system must be partitioned to prevent conflicts (for example, interference in wireless communication, contention in parallel processing, or collisions in scheduling); proper coloring directly models this requirement. Second, the partitioned components must remain observable or controllable via a limited supervisory mechanism; classical domination captures such coverage, but it is typically local and adjacency-based. Dominated coloring strengthens the interaction between partitioning and supervision by requiring each color class to be dominated by at least one vertex, but this still abstracts away the role of communication distance and the fact that monitoring and coordination often occur along shortest routes. GDC consolidates these requirements: it enforces conflict-free partitioning while ensuring that every class can be supervised through a shortest-path backbone. In this way, GDC simultaneously encodes (i) separation of incompatible vertices, (ii) structured coverage of each class, and (iii) efficient reachability through geodesics.

The above synthesis becomes particularly compelling for networks whose topology is inherently modular, hierarchical, or redundant. Such architectures are naturally modeled by product graphs, which assemble large graphs from simpler components while preserving structural regularities. For instance, the lexicographic product captures hierarchical layering by replacing each vertex of a backbone by a local module; vertex corona constructions model hub-cluster designs by attaching local components to backbone vertices; and Hamiltonian edge corona variants encode redundancy along selected edges to improve robustness. In these composite settings, shortest-path geometry can change dramatically under graph composition, and the interaction between coloring constraints and distance-based domination becomes non-trivial. Consequently, studying GDC on product graphs not only yields new structural bounds and extremal phenomena but also provides principled guidance for the design of scalable and fault-tolerant architectures where partitioning and shortest-path-based supervision must be achieved simultaneously.

4 Some known results

Observation 4.1. [25] For any graph G, .

Observation 4.2. [25] If G contains a geodesic that dominates V(G), then .

Theorem 4.3. [25] If G is a graph of order n with a supreme geodesic λ, where m = |I(λ)|/2, then .

Theorem 4.4. [25] If G is a graph on order n with a supreme geodesic λ, and H is the induced subgraph on V(I(λ)) with minimum chromatic number among all geodesics in G, where , then .

5 Main results

Observation 5.1. For any graph G of order n, .

Theorem 5.2. For any graphs G and H, .

Proof. Let (u, v) be an edge in G+H such that u∈G and v∈H. Since G+H is a join graph, V(H)⊆N(u) and V(G)⊆N(v). Thus, the geodesic between u and v dominates V(G+H). By Observation 4.2, . It is a well-known fact that χ(G+H) = χ(G)+χ(H). Therefore, .

Theorem 5.3. For any graphs G and H, .

Proof. Let c:V(G∪H) → [k] be a GDC of G∪H. Consider the restriction of c to V(G). If two vertices are adjacent in G, they are also adjacent in G∪H. Since c is proper on G∪H, such vertices receive distinct colors. Hence, the restriction c to V(G) is proper on G. For the geodesic domination property, note that it is verified only with respect to geodesics lying entirely inside G. Thus, the restriction c to V(G) is a valid GDC of G. By the same reasoning, the restriction of c to V(H) is a GDC of H. Hence, and , so that . Therefore, .

Remark 5.4. For any vertex-disjoint graphs G and H,

Theorem 5.5. For any graphs G and H,

Proof. Let c_G:V(G) → [p] and c_H:V(H) → [q] be a GDCs of G and H, respectively, where and . Since G∩H is a subgraph of both G and H, the restriction of c_G (or c_H) to V(G∩H) induces a proper coloring of G∩H. Consider the restriction of c_G to V(G∩H), denoted by c. For each color class C(i) of G∩H under c, where i∈[p], there exists a geodesic λ in G such that C(i)⊆I(λ), since c_G is a GDC of G. Note that every subpath of a geodesic is itself a geodesic between its endpoints. Hence, the subpath of λ lying entirely in G∩H, say λ′, is a geodesic in G∩H satisfying C(i)⊆I(λ′). Consequently, c is a GDC of G∩H. Hence, . By symmetry, considering the restriction of c_H to V(G∩H) yields . Therefore, .

Remark 5.6. For any subgraph H of G, .

The proof of the following theorem is omitted, as it is similar to that of Observation 5.1.

Theorem 5.7. For any graphs G and H with n and m vertices respectively,

.

Theorem 5.8. For any graphs G and H, if G contains a geodesic λ whose closure dominates V(G), then .

Proof. Let λ be a geodesic in G such that I(λ) = V(G). For any vertex (x, h)∈V(G•H), since I(λ) = V(G), there exists a vertex y∈V(λ) with either y = x or (y, x)∈E(G). In either case, every vertex in the copy of H corresponding to x is adjacent to some vertex associated with y in G•H. Hence, all copies of H are dominated by vertices corresponding to λ, implying I(λ) = V(G•H). Therefore, by Observation 4.2, .

Theorem 5.10. For any graphs G and H of order n and m, respectively, .

Proof. Let c_G:V(G) → [p] and c_H:V(H) → [q] be a GDCs of G and H, respectively, where and . It follows from the construction of G•H that there are m copies of G and n copies of H. For each g∈V(G) and h∈V(H), let H_g and G_h denote the induced subgraphs of G•H on the vertex sets {(g, h_j):1 ≤ j ≤ m} and {(g_i, h):1 ≤ i ≤ n}, respectively. Clearly, H_g≅H and G_h≅G. Let c:V(G•H) → [pq]. For 1 ≤ i ≤ p and for each g∈C(i), assign c((g, h_j)) = c_H(h_j)+q(i−1), 1 ≤ j ≤ m. Observe that for any g∈V(G), the restriction of c to H_g induces a proper coloring of H_g, since c_H is a proper coloring of H. Furthermore, if g and g′ are adjacent vertices in G, then the color sets assigned to H_g and are disjoint by construction. Thus, we have c(x)≠c(y) for all x∈V(H_g) and . As a result, c is a proper coloring of G•H. It suffices to verify that each color class of G•H is dominated by at least one geodesic. For 1 ≤ i ≤ p, let S_i = {(g, h_j):g∈C(i)and1 ≤ j ≤ m}⊆V(G•H). Since c_G is a GDC of G, there exists a geodesic λ_i in G that dominates the color class C(i). This implies that the corresponding geodesic in G•H dominates the set S_i. As a result, c is a GDC of G•H. Therefore, .

We now verify that the bounds given in Theorem 5.9 are sharp:Upper Bound: Consider two paths P_n and P_m. It is well-known that [25], . Furthermore, it is easy to observe that . By Theorem 5.9, we have . Therefore, Thus, the upper bound in Theorem 5.9 is attained, confirming its sharpness.

Theorem 5.10. For any cycle G of order n>5 and any complete graph H of order m, let . Then .

Proof. Let c:V(G•H) → [k] be a GDC of G•H and λ be a supreme geodesic in G•H. It is easy to observe that the length of λ is d(G•H) and |I(λ)| = m(d(G•H)+3). Since G is a cycle and H is a complete graph, it follows that d(G•H) = d(G), and any two vertices belonging to the same row or to adjacent rows in G•H must receive distinct colors under c. This implies that , ∀i∈[k]. Therefore, . Let as defined in Algorithm 1. See Figures 1, 2.

Algorithm 1

GDC of lexicographic product.

Figure 1

Figure 2

Proof of correctness: We prove that Algorithm 1 produces a GDC of G•H = C_n•K_m. First, we show that the coloring produced by Algorithm 1 is proper. The algorithm assigns labels to the vertices v_{(i, j)} column by column, in such a way that within every fixed column j, vertices are labeled alternately along the cycle C_n. The coloring is then defined by . Hence, any two vertices receiving the same color belong to the same block of at most r consecutive labels. By the labeling pattern in Algorithm 1, vertices with the same color are pairwise separated by at least one intermediate vertex along each copy of C_n. Moreover, since H = K_m, every two vertices lying in the same row are adjacent, and Algorithm 1 assigns distinct labels, hence distinct colors, to all vertices in the same row. Therefore, no two adjacent vertices of C_n•K_m receive the same color. Therefore, c is a proper coloring of G•H. It suffices to verify that each color class of G•H is dominated by at least one geodesic. Let C(i) be a color class such that |C(i)| = r for some i∈[k]. By Algorithm 1, there exists a cycle C (a copy of G) in G•H that contains all the vertices of C(i). Let (u₁, u₂, …, u_r) denote the increasing sequence of vertices of C(i) under the labeling specified in Algorithm 1. Observe that in this sequence the vertices of C(i) appear alternately along C. Since the vertices of C(i) are placed alternately on C, it follows that in C, d(u₁, u_r) = 2r−2. Let x and y be two vertices on C such that x∈N(u₁)∩N(u₂)∩V(C) and y∈N(u_r−1)∩N(u_r)∩V(C). Let P be the path from x to y in C that passes through the vertices u₂, u₃, …, u_r−1. Then d(x, y) = 2r−4 ≤ d(G), which implies that P is a shortest path between x and y in C. Since every geodesic in C is also a geodesic in G•H, P is a geodesic in G•H. By the choice of x and y, it follows that C(i)⊆I(P). Thus, the geodesic P dominates the color class C(i). By similar reasoning, for any color class C(i) with |C(i)| ≤ r, there exists a geodesic in G•H that dominates it. As a result, c is a GDC of G•H. Therefore, .

Theorem 5.11. For any cycles G and H of orders n>5 and m respectively, with n<m, let and . Then

Proof. Consider the case when m is even. Let c:V(G•H) → [k] be a GDC of G•H and λ be a supreme geodesic in G•H. It is easy to observe that the length of λ is d(G•H) and |I(λ)| = m(d(G•H)+3). Let S be the induced subgraph on the set of vertices I(λ). Since S contains a complete graph on 4 vertices, χ(S)≥4. This implies that , ∀i∈[k]. Moreover, there is no geodesic in G such that the subgraph induced by its closure cannot be colored with fewer than four colors. Therefore, by Theorem 4.4, we have . Let as defined in Algorithm 2. See Figure 3.

Algorithm 2

GDC of lexicographic product.

Figure 3

Proof of correctness: We prove that Algorithm 2 produces a GDC of G•H = C_n•C_m. First, we show that the coloring produced by Algorithm 2 is proper. The algorithm assigns labels to the vertices of G•H by first labeling the columns j = 1, 2, m, and then extending the labeling to the remaining columns by copying the labels from the column indexed by j−2. The coloring is then defined by . Hence, any two vertices receiving the same color belong to the same block of at most s consecutive labels. By the labeling pattern in Algorithm 2, vertices assigned the same color occur only in rows of the same parity and in columns whose indices differ by two. Consequently, any two vertices having the same color are separated by at least one intermediate vertex in every copy of G as well as in every copy of H, and therefore they cannot be adjacent. In particular, no two vertices of the same color lie in consecutive rows, and no two vertices of the same color lie in consecutive columns. Since adjacency in C_n•C_m arises either from vertices lying in consecutive rows or from vertices lying in the same row and in consecutive columns, it follows that no two adjacent vertices of C_n•C_m receive the same color. Therefore, c is a proper coloring of G•H. It suffices to verify that each color class of G•H is dominated by at least one geodesic. Let C(i) be a color class such that |C(i)| = r for some i∈[k]. By Algorithm 2, there exists a cycle C (a copy of G) in G•H that contains vertices of C(i), and these t vertices are located either entirely within a single column of G•H or within two consecutive columns. Let (u₁, u₂, …, u_t) denote the increasing sequence of those t vertices of C(i) that lie on the cycle C, under the labeling specified in Algorithm 2. Observe that in this sequence the vertices of C(i) appear alternately along C. Since the vertices of C(i) are placed alternately on C, it follows that in C, d(u₁, u_t) = 2t−2. Let x and y be two vertices on C such that x∈N(u₁)∩N(u₂)∩V(C) and y∈N(u_t−1)∩N(u_t)∩V(C). Let P be the path from x to y in C that passes through the vertices u₂, u₃, …, u_t−1. Then d(x, y) = 2t−4 ≤ d(G), which implies that P is a shortest path between x and y in C. Since every shortest path in C is also a shortest path in G•H, P is a geodesic in G•H. By the choice of x and y, the vertices u₁, u₂, …, u_t lie in I(P). Moreover, the remaining r−t vertices of C(i) lie outside the cycle C, each lying in the same row as some vertex of C in G•H. By construction of G•H, each of these vertices is adjacent to some vertex on the path P, and hence belongs to I(P). Thus, the geodesic P dominates the color class C(i). By similar reasoning, for any color class C(i) with |C(i)| ≤ r, there exists a geodesic in G•H that dominates it. As a result, c is a GDC of G•H. Therefore, . The case where m is odd can be dealt with similarly by Algorithm 3. See Figure 4.

Algorithm 3

GDC of lexicographic product.

Figure 4

Corollary 5.12. For any complete graph K_n and any path P_m, .

Corollary 5.13. For any complete graph K_n and any cycle C_m,

Theorem 5.14. For any graphs G and H of order n and m≥2, respectively, .

Proof. Let c_H:V(H) → [p] be a GDC of H, where p = χ(H). For each g_i∈V(G), let H_gi denote the copy of H corresponding to g_i, where 1 ≤ i ≤ n. Consider the coloring c:V(G°H) → [np] by c(x) = c_H(x)+p(i−1), for eachx∈V(H_gi). Since c_H is a proper coloring of H, its restriction to each copy H_gi is proper, and distinct copies of H receive disjoint color sets. To extend the coloring to the vertices of G, let M be a maximum matching of G. For each edge (g_i, g_j)∈M, assign c(g_i) = c(x), for somex∈V(H_gj) and c(g_j) = c(y), for somey∈V(H_gi). For every unmatched vertex g_i∈V(G)\V(M), select a neighbor g_j∈V(M) such that (g_j, g_k)∈M for some g_k∈V(G). Assign to g_i a color from H_gj distinct from the color of g_k. Such a choice is always possible since m≥2. We now verify that c is a proper coloring of G°H. For each copy H_gi, the coloring c is proper since it is induced by c_H, and different copies use disjoint sets of colors. On G, vertices matched in M receive distinct colors. Each unmatched vertex is assigned a color different from that of its matched neighbor, and any two unmatched vertices that share a color must be non-adjacent; otherwise, they would have been included in M. Thus, the restriction of c to G is proper, and hence c is a proper coloring of G°H. Next, we confirm that c is a GDC of G°H. For each g_i∈V(G), the subgraph induced by H_gi is isomorphic to H and colored using c_H shifted by p(i−1). Since c_H is a GDC of H, every color class within H_gi is dominated by a geodesic. Thus, the restriction of c to each H_gi is also geodesic dominated. Moreover, g_i is adjacent to all vertices in H_gi, which collectively use the color set [p(i−1)+1, pi]. If some vertex g_j∈V(G) is assigned a color from H_gi, then g_i is also adjacent to g_j since (g_i, g_j)∈E(G). Thus, g_i itself dominates the entire color set [p(i−1)+1, pi]. As a result, c is a GDC of G°H. Therefore, .

Theorem 5.15. For any complete graphs G and H of order n≥3 and m ≤ 2, respectively, .

Proof. Let c:V(G°H) → [k] be a GDC of G°H. Since G°H contains a complete graph on n vertices, χ(G°H)≥n. Hence, . Let c:V(G°H) → [n]. See Figure 5B. First, assign to each vertex of G a unique color from [n]. Second, consider Hamiltonian cycle 𝒞 in G. Then, for every u∈V(𝒞), color the vertices of H_u using colors from the set {c(x), c(y)} where x and y belongs to N(u)∩V(𝒞) in G°H. Clearly, c is a proper coloring of G°H. It suffices to verify that each color class C(i), for every i∈[n] is dominated by at least one geodesic. We consider two distinct cases. In the first case, there are two vertices of two copies of H shares a color from [n]. Now the geodesic between two vertices of V(G°H)\V(G) that are colored with i, dominates the color class C(i), i∈[n]. In the second case, there is only one vertex from a copy of H shares a color from [n]. The geodesic between the vertices x and y that are colored with i dominates the color class C(i) where x and y belongs to a copy of H and V(G), respectively. This is true for every i∈[n]. Therefore, c is a GDC of G°H.

Figure 5

Theorem 5.16. For any complete graphs G and H of order n≥3 and m≥3, respectively, .

Proof. Let c:V(G°H) → [k] be a GDC of G°H. Since G°H contains a complete graph on n vertices, χ(G°H)≥n. Hence, . Moreover, these n colors are used to color the vertices of G in G°H. Now we assert that a color used in V(G) can be shared with at most two vertices of V(G°H)\V(G) such that they belong to distinct copies of H. Assume the contrary, suppose a color that is used in V(G) is shared with three vertices, say x, y, and z of V(G°H)\V(G). Clearly, these three vertices belong to distinct copies of H. Since c is a GDC of G°H, there exists a geodesic in G°H that dominates the vertices x, y, and z. This implies that there exists a geodesic in G°H that dominates three copies of H. However, in G°H, the geodesic between any two vertices dominates at most two copies of H, because both G and H are complete graphs. Thus, no geodesic in G°H can dominate three distinct copies of H. This is a contradiction. Hence, a color used in V(G) can be shared with no more than two vertices of V(G°H)\V(G) such that they belong to distinct copies of H. Now to color the remaining m−2 vertices from each copy of H, we additionally need at least n(m−2)/2 colors since a geodesic in G°H dominates at most two vertices of V(G°H)\V(G) such that they belong to distinct copies of H. Therefore, . Let c:V(G°H) → [nm/2] as defined in Algorithm 4. See Figure 5A.

Algorithm 4

GDC of °.

Proof of correctness: We prove that Algorithm 4 produces a GDC of G°H = K_n°K_m, where n≥3 and m≥3. First, we show that the coloring produced by Algorithm 4 is proper. The algorithm assigns distinct colors to the vertices v₁, v₂, …, v_n of G, and then assigns colors to the vertices belonging to the copies of H according to the proposed coloring scheme in Algorithm 4. By construction, each color class contains at most two vertices, and whenever two vertices receive the same color, they belong to distinct copies of H. Now, in the corona product K_n°K_m, adjacency arises in exactly three ways: either between two vertices of G, or between two vertices belonging to the same copy of H, or between a vertex v_i of G and a vertex of the corresponding copy H_i. Since G = K_n, the vertices v₁, v₂, …, v_n are pairwise adjacent, and Algorithm 4 assigns them distinct colors. Further, each copy H_i is a complete graph, so any two vertices within the same copy are adjacent; however, by the coloring rule of Algorithm 4, no two vertices in the same copy of H receive the same color. Also, no vertex v_i and no vertex in its corresponding copy H_i receive the same color. Therefore, any two vertices assigned the same color must belong to distinct copies of H. Since there are no edges between vertices belonging to different copies of H, such vertices are non-adjacent. Hence, no two adjacent vertices of G°H receive the same color. Therefore, c is a proper coloring of G°H. It suffices to verify that each color class of G°H is dominated by at least one geodesic. Now, the geodesic between two vertices of V(G°H)\V(G) colored with i dominates the color class C(i), i∈[nm/2]. This is true for every i∈[nm/2]. As a result, c is a GDC of G°H. Therefore, .

Proposition 5.17. For any graphs G and H of order n and m, respectively, .

Theorem 5.18. For any complete graphs G and H of order n and m, respectively,

Proof. Let c:V(G♦H) → [k]. Consider the case when n≥m+2. In this case, χ(G♦H)≥χ(G)≥n since m ≤ n−2. Consider the case when n<m+2. In this case, χ(G♦H)≥χ(H)≥m since m≥n−1. Further, every copy of H in G♦H is joined to two adjacent vertices of G. Hence, χ(G♦H)≥m+2. Let 𝒞 be a Hamiltonian cycle in G, and for each edge (u, v)∈E(𝒞), let H_uv denote the copy of H associated with (u, v).

Case 1: n≥m+2. Consider the coloring c:V(G♦H) → [n] by assigning pairwise distinct colors from [n] to the vertices of G. Since every vertex of H_uv is adjacent to both u and v, its m vertices are assigned distinct colors from [n]\{c(u), c(v)}, which is possible because n≥m+2. Thus, each H_uv is properly colored, and c yields a proper coloring of G♦H. See Figure 6A.

Figure 6

Case 2: n<m+2. Consider the coloring c:V(G♦H) → [m+2] analogously, assigning pairwise distinct colors from [n] to V(G). For each edge (u, v)∈E(𝒞), the corresponding H_uv is colored with distinct colors from [m+2]\{c(u), c(v)}, which is possible beacuse |[m+2]\{c(u), c(v)}| = m. Hence, each H_uv is properly colored, and c yields a proper coloring of G♦H. See Figure 6B.

Theorem 5.19. For any Hamiltonian graph G of order n and any graph H of order m≥2, .

Proof. Let c_H:V(H) → [p] be a GDC of H, where p = χ(H). Let V(G♦H) = {g₁, g₂, …, g_n} and let 𝒞 be a Hamiltonian cycle in G with edge set E(𝒞) = {e_i = (g_i, g_i+1), i∈[n]}, with cyclic indexing on the vertices (i.e., g_n+1 = g₁). For each edge e_i∈E(𝒞), let H_ei denote the corresponding copy of H attached to e_i, 1 ≤ i ≤ n. Consider the coloring c:V(G♦H) → [np] by c(x) = c_H(x)+p(i−1), for eachx∈V(H_ei). Since c_H is a proper coloring of H, its restriction to each copy H_ei is proper, and distinct copies of H receive disjoint color sets. To extend the coloring to V(G), for each g_i∈V(G) assign c(g_i) = c(x), for somex∈V(H_ei+1). We now verify that c is a proper coloring of G♦H. For each copy H_ei, the coloring c is proper since it is induced by c_H, and different copies use disjoint sets of colors. For the vertices of G, each g_i receives its color from H_ei+1, which is distinct from its incident copy H_ei. Moreover, g_i is not adjacent to any vertex of H_ei+1, ensuring that no conflict arises. Thus, c restricted to G is proper, and hence c is a proper coloring of G♦H. Next, we confirm thatc is a GDC of G♦H. For each edge (g_i, g_i+1)∈E(𝒞), the subgraph induced by H_ei is isomorphic to H and is colored using c_H shifted by p(i−1). Observe that the geodesic between g_i and g_i+1 dominates the color classes C(p(i−1)+1), C(p(i−1)+2), …, C(pi). As a result, c is a GDC of G♦H. Therefore, .

Theorem 5.20. For any complete graphs G and H of order n≥3 and m, respectively,

Proof. Case 1: n ≤ 5 and m ≤ 3, or n ≤ 4 and m≥4, or n≥6 and m ≤ 4.By Observation 4.1, . Let c:V(G♦H) → [p] as defined in Algorithm 5, where p = χ(G♦H). See Figure 6A.

Algorithm 5

GDC of the Hamiltonian Edge Corona ♦.

Proof of correctness: We prove that the coloring produced by Algorithm 5 is proper. The algorithm first assigns distinct colors to the vertices of G = K_n, namely c(v_i) = i, 1 ≤ i ≤ n. Hence, since G is complete, no two adjacent vertices of G receive the same color. Now consider a copy H_uv≅K_m corresponding to an edge (u, v) of the Hamiltonian cycle C in G. Since every vertex of H_uv is adjacent to both u and v, Algorithm 5 assigns to each chosen vertex of H_uv a color from {1, 2, …, n}\{c(u), c(v)}, thereby avoiding conflict with the endpoints u and v. Moreover, because H_uv is a complete graph, any two vertices of H_uv are adjacent, and hence the algorithm assigns distinct colors to vertices lying in the same copy of H. For the remaining vertices, Algorithm 5 assigns a new color to each group formed by taking one vertex from distinct copies of H along the Hamiltonian cycle C. Since there are no edges between vertices belonging to different copies of H, any two vertices receiving the same new color are non-adjacent. Therefore, no two adjacent vertices of G◇H receive the same color. Hence, the coloring produced by Algorithm 5 is proper. We now verify that c is a GDC of G♦H. For n = 3, every geodesic between two vertices of G dominates all color classes of G♦H. For n = 4, any geodesic between two vertices of a Hamiltonian cycle 𝒞 in G at distance two dominates all color classes. For n = 5, let i∈[p] be the color assigned to a vertex u∈V(G). Let x and y be vertices of a Hamiltonian cycle 𝒞 in G such that d(u, x) = d(u, y) = 2 in 𝒞. Then the geodesic between x and y in G dominates the entire color class C(i), since m ≤ 3. The same argument applies to every i∈[p]. For n≥6, the reasoning is analogous and is therefore omitted. As a result, c is a GDC of G♦H. Therefore, . Case 2: n = 5 and m≥4.

Let c:V(G♦H) → [k] be a GDC of G♦H. we assert that any color from [k] can be assigned to at most four vertices of V(G♦H). Suppose, to the contrary, that a color is shared by five vertices. Now we have the following two subcases: in Subcase (a), all five vertices belong to distinct copies of H, and since each edge of G dominates at most four copies of H, no geodesic can dominate five vertices from five distinct copies, leading to a contradiction; in Subcase (b), one vertex, say u, belongs to V(G) and the remaining four, say a, b, c, d, belong to V(G♦H)\V(G). Since a, b, c and d belong to distinct copies of H, and u forbids its two adjacent copies in a Hamiltonian cycle 𝒞 of G, at most three copies of H can share the same color, which again leads to a contradiction. Hence, each color can be assigned to at most four vertices of V(G♦H). Therefore, . We now prove the upper bound. Let c:V(G♦H) → [n(m+1)/4] as defined in Algorithm 6. See Figure 6C.

Algorithm 6

GDC of the Hamiltonian Edge Corona ♦.

Proof of correctness: We prove that the coloring produced by Algorithm 6 is proper. The algorithm first assigns distinct colors to the vertices of G = K₅, namely c(v_i) = i, 1 ≤ i ≤ 5. Since G is complete, this yields a proper coloring on the vertex set V(G). Next, for each copy H_uv≅K_m corresponding to an edge (u, v) of the Hamiltonian cycle C, Algorithm 6 chooses three vertices and assigns them distinct colors from {1, 2, …, 5}\{c(u), c(v)}. Thus, none of these vertices receives the same color as either endpoint u or v, and since H_uv is complete, no two adjacent vertices in H_uv receive the same color. The remaining vertices are partitioned into groups of size at most four, each group consisting of one vertex taken from four consecutive copies of H along the cycle C. Each such group is assigned a new color. Since the vertices of a given group belong to distinct copies of H, and there are no edges between different copies, any two vertices receiving the same new color are non-adjacent. Therefore, no two adjacent vertices of G◇H receive the same color. Hence, the coloring produced by Algorithm 6 is proper. We now verify that c is a GDC of G♦H. Now consider two subcases based on how the color is assigned to vertices:

Subcase (a): A color is assigned to four vertices of V(G♦H). Let i∈[⌈n(m+1)/4⌉] be such a color assigned to vertices u, v, w, x∈V(G♦H). Regardless of whether three of these vertices belong to consecutive copies of H and the fourth to V(G), or all four belong to four consecutive copies of H, by construction there exist vertices y∈N(u)∩N(v) and z∈N(w)∩N(x) such that the geodesic between y and z in G dominates the color class C(i).

Subcase (b): A color is assigned to at most three vertices of V(G♦H)\V(G). Let i∈[⌈n(m+1)/4⌉] be such a color assigned to vertices u, v, w∈V(G♦H)\V(G). Since the reasoning is analogous to Case (i), the color class C(i) is dominated by a suitable geodesic in G. Similarly, if a color is assigned to two or one vertex, the same argument applies. Since this argument holds for every color i∈[⌈n(m+1)/4⌉], it follows that c is a GDC of G♦H. Therefore, .

Case 3: n≥6 and m≥5.

Let c:V(G♦H) → [k] be a GDC of G♦H. See Algorithm 7 and Figure 6D. Now, the proof that c is a GDC of G♦H is similar to that of Case 2 and hence omitted. Therefore, .

Algorithm 7

GDC of the Hamiltonian Edge Corona ♦.

6 Conclusion

In this paper, we studied GDC in product graphs and investigated how graph composition influences the geodesic dominated chromatic number. By analyzing standard graph products, lexicographic, vertex corona, and Hamiltonian edge corona, we provided exact values and general bounds for the geodesic dominated chromatic number for several graph classes. Our results show how the structural properties of product graphs influence geodesic domination and coloring, highlighting the role of supreme geodesics and induced subgraphs in determining sharp bounds. These findings contribute to the theoretical understanding of geodesic-based coloring and offer practical insights for network design, including fault tolerance, load balancing, and efficient routing.

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