Odd edge coloring of graphs

Abstract

An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and each color c, the vertex v uses the color c an odd number of times or does not use it at all. In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3,2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved. We also pose several open problems.

Odd 4‐edge‐colorability of graphs

“…As defined in , the reduction red( G ) of a loopless graph G is a spanning subgraph obtained by reducing each bouquet to size either 1 or 2 depending on whether is odd or even (c.f. Fig.…”

“…and is of the same type as G . The inequality below can be easily shown for all loopless graphs (see ). …”

Odd edge-colorability of subcubic graphs

“…odd) number of parallel edges. Then G is a Shannon triangle of type (p, q, r), and it holds that χ o (G) = p + q + r. The following result was proven in [4]. Theorem 1.2.…”

“…(1,1,1) (2,1,1) (2,2,1) (2,2,2) As defined in [4], a Shannon triangle is a loopless graph on three pairwise adjacent vertices. Observe that for any Shannon triangle, as a direct consequence of the handshake lemma, the edge set of every odd subgraph is fully contained in a single bouquet.…”

“…In [4] it was conjectured that the problem of determining whether an arbitrary loopless graph G is odd 2-edge-colorable is NP-hard. Perhaps for some values of ∆(G) beyond 3 this is still decidable in polynomial time.…”