Odd edge-colorability of subcubic graphs

Abstract

An edge-coloring of a graph G is said to be odd if for each vertex v of G and each color c, the vertex v either uses the color c an odd number of times or does not use it at all. The minimum number of colors needed for an odd edge-coloring of G is the odd chromatic index χ o (G). These notions were introduced by Pyber in [7], who showed that 4 colors suffice for an odd edge-coloring of any simple graph. In this paper, we consider loopless subcubic graphs, and give a complete characterization in terms of the value of their odd chromatic index.

Remarks on odd colorings of graphs

“…An odd coloring of hypergraph H is a coloring such that for every edge e ∈ E(H) there is a color c with an odd number of vertices of e colored by c. Particular features of the same notion notion (under the name weak-parity coloring) have been considered by Fabrici and Göring [10] (in regard to face-hypergraphs of planar graphs) and also by Bunde et al [6] (in regard to coloring of graphs with respect to paths, i.e., path-hypergraphs). For various edge colorings of graphs with parity condition required at the vertices we refer the reader to [4,8,12,13,14,15,16].…”

Colorings with neighborhood parity condition

“…An odd coloring of hypergraph H is a coloring such that for every edge e ∈ E(H) there is a color c with an odd number of vertices of e colored by c. Particular features of the same notion (under the name weak-parity coloring) have been considered by Fabrici and Göring [10] (in regard to face-hypergraphs of planar graphs) and also by Bunde et al [4] (in regard to coloring of graphs with respect to paths, i.e., path-hypergraphs). For various edge colorings of graphs with parity condition required at the vertices we refer the reader to [2,6,12,14,15,17,18].…”

Odd 4‐edge‐colorability of graphs

“…If G admits odd edge‐colorings, the odd chromatic index is the minimum integer k for which G is odd k ‐edge‐colorable. A characterization of subcubic graphs in terms of is given in . An obvious necessary and sufficient condition for odd edge‐colorability of G is the absence of vertices incident only to loops.…”