Odd 4‐edge‐colorability of graphs

Abstract

An odd k‐edge‐coloring of a graph G is a (not necessarily proper) edge‐coloring with at most k colors such that each nonempty color class induces a graph in which every vertex is of odd degree. Pyber (1991) showed that every simple graph is odd 4‐edge‐colorable, and Lužar et al. (2015) showed that connected loopless graphs are odd 5‐edge‐colorable, with one particular exception that is odd 6‐edge‐colorable. In this article, we prove that connected loopless graphs are odd 4‐edge‐colorable, with two particular exceptions that are respectively odd 5‐ and odd 6‐edge‐colorable. Moreover, a color class can be reduced to a size at most 2.

The odd chromatic index of almost all graphs

“…We established the odd chromatic index of almost all graphs, and proved that it depends only on the parity of the number of vertices of the graph. It might also be interesting to prove a result similar to Theorem 1 for multigraphs, for which the odd chromatic index can go up to 6 [Lužar et al 2014, Petruševski 2018. In another direction, we may consider decompositions in graphs in which every vertex has degree 1 modulo k. In this case, the best known upper bound to the modulo k chromatic index is 5k 2 log k [Scott 1997].…”

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].…”

Coverability of graph by three odd subgraphs

“…As Theorem 2.5 extends Theorem 2.2, the principal aim of this paper is an analogous extension of Theorem 2.3 (conjectured in [9]). Our characterization of odd 3-edge-coverability requires another notion from [6].…”

“…yields the same tight upper bound on G cov ( ) o (with the same characterization of equality). In [9], Theorem 2.4 was further improved into the following characterization of odd 4-edge-colorability:…”