A new vertex coloring heuristic and corresponding chromatic number
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One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy (First-Fit) coloring and color-dominating colorings of graphs are two well-known such techniques. The color-dominating colorings are also known and commonly referred as b-colorings. But these two topics have been studied separately in graph theory. We introduce a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property. We first prove that the vertices of every graph G can be effectively colored using color classes say C_1, …, C_k such that (i) for any two colors i and j with 1≤ i< j ≤ k, any vertex of color j is adjacent to a vertex of color i, (ii) there exists a set {u_1, …, u_k} of vertices of G such that u_j∈ C_j for any j∈{1, …, k} and u_k is adjacent to u_j for each 1≤ j ≤ k with j≠ k, and (iii) for each i and j with i≠ j, the vertex u_j has a neighbor in C_i. This provides a new vertex coloring heuristic which improves both Grundy and color-dominating colorings. Denote by z(G) the maximum number of colors used in any proper vertex coloring satisfying the above properties. The z(G) quantifies the worst-case behavior of the heuristic. We prove the existence of {G_n}_n≥ 1 such that min{Γ(G_n), b(G_n)}→∞ but z(G_n)≤ 3 for each n. For each positive integer t we construct a family of finitely many colored graphs 𝒟_t satisfying the property that if z(G)≥ t for a graph G then G contains an element from 𝒟_t as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for z(G).
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