Sudoku number of graphs

We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let G=(V,E) be a graph of order n with chromatic number χ(G)=k and let S⊆V. Let C0 be a k-coloring of the induced subgraph G[S]. The coloring C0 is called an extendable coloring if C0 can be extended to a k-coloring of G. We say that C0 is a Sudoku coloring of G if C0 can be uniquely extended to a k-coloring of G. The smallest order of such an induced subgraph G[S] of G which admits a Sudoku coloring is called the Sudoku number of G and is denoted by sn(G). In this paper we initiate a study of this parameter. We first show that this parameter is related to list coloring of graphs. In Section 2, basic properties of Sudoku coloring that are related to color dominating vertices, chromatic numbers and degree of vertices, are given. Particularly, we obtained necessary conditions for C0 being extendable, and for C0 being a Sudoku coloring. In Section 3, we determined the Sudoku number of various families of graphs. Particularly, we showed that a connected graph G has sn(G) = 1 if and only if G is bipartite. Consequently, every tree T has sn(T) = 1. We also proved that sn(G)=|V(G)|−1 if and only if G = Kn. Moreover, a graph G with small chromatic number may have arbitrarily large Sudoku number. In Section 4, we proved that extendable partial coloring problem is NP-complete. Extendable coloring and Sudoku coloring are nice tools for providing a k-coloring of G.