Sudoku number of graphs

Abstract

AbstractWe introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let [Formula: see text] be a graph of order n with chromatic number [Formula: see text] and let [Formula: see text] Let [Formula: see text] be a k-coloring of the induced subgraph [Formula: see text] The coloring [Formula: see text] is called an extendable coloring if [Formula: see text] can be extended to a k-coloring of G. We say that [Formula: see text] is a Sudoku coloring of G if [Formula: see text] can be uniquely extended to a k-coloring of G. The smallest order of such an induced subgraph [Formula: see text] of G which admits a Sudoku coloring is called the Sudoku number of G and is denoted by [Formula: see text] 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 [Formula: see text] being extendable, and for [Formula: see text] 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 [Formula: see text] 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.

Keywords

• Chromatic number
• extendable coloring
• Sudoku coloring
• 05C78
• 05C69