Notes on weak-odd edge colorings of digraphs

Abstract

A weak-odd edge coloring of a general digraph D is a (not necessarily proper) coloring of its edges such that for each vertex v ∈ V (D) at least one color c satisfies the following conditions: if d − D (v) > 0 then c appears an odd number of times on the incoming edges at v; and if d + D (v) > 0 then c appears an odd number of times on the outgoing edges at v. The minimum number of colors sufficient for a weak-odd edge coloring of D is the weak-odd chromatic index, denoted χ ′ wo (D). It is known that χ ′ wo (D) ≤ 3 for every digraph D, and that the bound is sharp. In this article we show that the weak-odd chromatic index can be determined in polynomial time. Restricting to edge colorings of D with at most two colors, the minimum number of vertices v ∈ V (D) for which no color c satisfies the above conditions is the defect of D, denoted def(D). Surprisingly, it turns out that the problem of determining the defect of digraphs is (polynomially) equivalent to the problem of finding the matching number of simple graphs. Moreover, we characterize the classes of associated digraphs and tournaments in terms of the weak-odd chromatic index and the defect.

Weak-odd chromatic index of special digraph classes

“…They believed that a descriptive characterization similar to graphs is impossible for all digraphs and they believed that deciding the exact value of χ ′ wo (D) is NP-hard. In [6], the authors showed a necessary and sufficient condition for digraphs to be weak-odd 2-edge colorable, and thus χ ′ wo (D) can be determined in polynomial time. When limit to 2 colors, use def(D) to denote the defect of D, the minimum number of vertices in D at which the condition ( − − → WO) is not satisfied.…”

“…When limit to 2 colors, use def(D) to denote the defect of D, the minimum number of vertices in D at which the condition ( − − → WO) is not satisfied. Hernández-Cruz, Petruevski, and Krekovski [6] proved that def(D) is related to the matching number of some graphs.…”

“…By extended tournaments we mean the digraph obtained from a tournament by blowing up some of its vertices into independent sets. Hernández-Cruz, Petruevski, and Krekovski [6] made a descriptive characterization for tournaments of the weak-odd chromatic index as follows.…”

“…Problem 1.2 ( [6]) Characterize the families of semicomplete digraphs, extended tournaments and multipartite tournaments in terms of their weak-odd chromatic index. Problem 1.3 ([6]) Characterize the defect in terms of the families of semicomplete digraphs, extended tournaments and multipartite tournaments when their defect are bounded.…”

“…Hernández-Cruz, Petruevski, and Krekovski [6] also started the study of weak-odd edge covering and showed the weak-odd 2-edge covering conditions for graphs. Then they asked the situation about digraphs.…”