A New Vertex Coloring Heuristic and Corresponding Chromatic Number

Abstract

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 $$C_1, \ldots , C_k$$ such that (i) for any two colors i and j with $$1\le i< j \le k$$ , any vertex of color j is adjacent to a vertex of color i, (ii) there exists a set $$\{u_1, \ldots , u_k\}$$ of vertices of G such that $$u_j\in C_j$$ for any $$j\in \{1, \ldots , k\}$$ and $$u_k$$ is adjacent to $$u_j$$ for each $$1\le j \le k$$ with $$j\not = k$$ , and (iii) for each i and j with $$i\not = j$$ , 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 $$\{G_n\}_{n\ge 1}$$ such that $$\min \{\Gamma (G_n), b(G_n)\} \rightarrow \infty $$ but $$z(G_n)\le 3$$ for each n. For each positive integer t we construct a family of finitely many colored graphs $${{\mathcal {D}}}_t$$ satisfying the property that if $$z(G)\ge t$$ for a graph G then G contains an element from $${{\mathcal {D}}}_t$$ as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for z(G).