Covering the edges of a graph by three odd subgraphs

Mátrai, Tamás

doi:10.1002/jgt.20170

Cited by 18 publications

(19 citation statements)

References 7 publications

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“…The following useful result appears in . For the sake of completeness, we present it with another proof.…”

Section: Preliminary Resultsmentioning

confidence: 92%

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Odd 4‐edge‐colorability of graphs

Petruševski¹

2017

Journal of Graph Theory

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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.

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“…The following useful result appears in . For the sake of completeness, we present it with another proof.…”

Section: Preliminary Resultsmentioning

confidence: 92%

“…Let us turn now to graphs of connectivity 1. As shown by Matrai , every connected loopless graph G with

χ_{o}^{'} false(G false) > 3

can be used to produce an infinite series of such graphs of connectivity 1. Namely, take an even number of copies of G and an additional vertex v .…”

Section: Related Resultsmentioning

confidence: 99%

Odd 4‐edge‐colorability of graphs

Petruševski¹

2017

Journal of Graph Theory

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“…As a notion, odd edge-covering was introduced in [5]. The scope of the mentioned work are all simple graphs, and the following result is proven.…”

Section: Odd Edge-coverabilitymentioning

confidence: 94%

“…The scope of the mentioned work are all simple graphs, and the following result is proven. Theorem 4.1 (Mátrai, 2006). For every simple graph G, it holds that cov o (G) ≤ 3.…”

Section: Odd Edge-coverabilitymentioning

confidence: 99%

Odd edge-colorability of subcubic graphs

2016

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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.

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“…In 2006, Mátrai [4] presented such a construction, taking an even number of copies of W 4 and an additional vertex v. Choosing an arbitrary edge from the wheel, he removed the same edge from every copy and connected its two end-vertices with v (see Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

Odd edge coloring of graphs

2015

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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.

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Covering the edges of a graph by three odd subgraphs

Abstract: We prove that any finite simple graph can be covered by three of its odd subgraphs, and we construct an infinite sequence of graphs where an edge-disjoint covering by three odd subgraphs is not possible. c

Cited by 18 publications

References 7 publications

Odd 4‐edge‐colorability of graphs

Odd 4‐edge‐colorability of graphs

Odd edge-colorability of subcubic graphs

Odd edge coloring of graphs

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