Abstract
A proper vertex coloring φ of graph G is said to be odd if for each non-isolated vertex x∈V(G) there exists a color c such that φ−1(c)∩N(x) is odd-sized. The minimum number of colors in any odd coloring of G, denoted χo(G), is the odd chromatic number. Odd colorings were recently introduced in Petruševski and Škrekovski (0000). Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to degenericity or maximum degree, and pose several questions and problems.
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