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.

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. 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. We prove that every simple planar graph admits an odd 9-coloring, and conjecture that 5 colors always suffice.

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.

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.

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.

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χ pcf (G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χ pcf (G) in terms of other graph parameters. In particular, we show that χ pcf (G) ≤ 5∆(G) 2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4 ≤ k ≤ 6. The paper concludes with few open problems.

No 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.