Remarks on odd colorings of graphs

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 χ o (G), is the odd chromatic number. Odd colorings were recently introduced in [M. Petruševski, R. Škrekovski: Colorings with neighborhood parity condition]. Here we discuss various basic properties of this new graph parameter, establish several upper bounds, several charatcterizatons, and pose some questions and problems.

A Note on Odd Colorings of 1-Planar Graphs

“…This is a contradiction. Corollary 3 below follows directly from Theorem 2 and the fact that every graph with no K 4 minor is 2-degenerate; it extends the result of Caro, Petruševski, and Škrekovski [3] on outerplanar graphs. The sharpness of Corollary 3 is witnessed by C 5 .…”

“…G of girth at least 7. Caro, Petruševski, and Škrekovski [3] studied various properties of the odd chromatic number; in particular, they proved the following facts: every outerplanar graph admits an odd 5-coloring; every graph of maximum degree three has an odd 4-coloring; and for every connected planar graph G, if |G| is even, or |G| is odd and G has a vertex of degree 2 or any odd degree, then χ o (G) ≤ 8 (a key step towards proving that 8 colors suffice for an odd coloring of any planar graph). It is worth noting that their proof of this key step relies on Theorem 4 in Aashtab, Akbari, Ghanbari, and Shidani [1], which itself relies on the Four-Color Theorem [2,12].…”

“…It is worth noting that their proof of this key step relies on Theorem 4 in Aashtab, Akbari, Ghanbari, and Shidani [1], which itself relies on the Four-Color Theorem [2,12]. Building on work of Caro, Petruševski, and Škrekovski [3], Petr and Portier [10] further proved that 8 colors suffice for all planar graphs; that is, every planar graph admits an odd 8-coloring.…”

“…If f is a 5-face, then f is incident to at most one 2-vertex (since every neighbor of each 2-vertex is a virtual 4-vertex and no two virtual 4-vertices are adjacent), so f sends 1 to v by (R1) and we are done. If f is a 7-face, then f is incident to at most two 2-vertices, so f sends v at least 3 2 , and we are done. If f is an 8 + -face, then f is incident to at most d(f ) 2 2-vertices, and…”

Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods

“…First, Cranston [8] established several results for sparse graphs, and as a corollary, he obtained bounds 6 and 5 for planar graphs of girth 6 and at least 7, respectively. Then, Caro et al [4] established the bound 8 for planar graphs with specific properties, and finally, Petr and Portier [19] proved the bound 8 for all planar graphs.…”