Partially normal 5-edge-colorings of cubic graphs

Abstract

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the five edges incident with an end of $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter ${\mathrm{\mu }}_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $\left|E\left(G\right)\right|-{\mathrm{\mu }}_3\left(G\right)$ (which is no less than $\frac{27}{35}\left|E\left(G\right)\right|$) edges are normal. This result improves on some earlier results of Bílková and Šámal.

Introduction

This paper focuses on Jaeger’s Petersen coloring conjecture [8], which states that every bridgeless cubic graph has a Petersen coloring. The truth of this conjecture implies the truth of some other conjectures, such as the Berge–Fulkerson conjecture and 5-cycle double cover conjecture (shortly, 5CDCC). There are several equivalent statements to the Petersen coloring conjecture, one of them states that every bridgeless cubic graph has a normal 5-edge-coloring. However, only few results on this conjecture are known. Here, we follow Šámal’s new approach [18] that might lead to a solution to Petersen coloring conjecture. For a given bridgeless cubic graph, we look for a proper 5-edge-coloring yielding as many normal edges as possible. In other words, we color the graph “as normal as possible” while the conjecture asserts that we can color the graph completely normal. The result of Bílková [1] targets some classes of cubic graphs and shows that, we can color a generalized prism so that $\frac{2}{3}$ of the edges are normal and we can color a cubic graph of large girth so that almost $\frac{1}{2}$ of the edges are normal. In this paper, we prove that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $\left|E\left(G\right)\right|-{\mu }_3\left(G\right)$ edges are normal. The parameter ${\mu }_3\left(G\right)$ can measure how far a cubic graph is from being 3-edge-colorable. The definition of ${\mu }_3\left(G\right)$ will be given in Section 1.3. By a result of Kaiser, Král and Norine in [12], it holds that ${\mu }_3\left(G\right)\le \frac{8}{35}\left|E\left(G\right)\right|$. Therefore, we can guarantee a proper 5-edge-coloring of $G$ with at least $\frac{27}{35}\left|E\left(G\right)\right|$ normal edges, which improves on these earlier results.

Given two cubic graphs $G$ and $H$, a mapping $\varphi :E\left(G\right)\to E\left(H\right)$ is an $H$-coloring of $G$ if any three mutually adjacent edges of $G$ are mapped to three mutually adjacent edges of $H$. The mapping $\varphi$ is called a Petersen-coloring if $H$ is the Petersen graph.

In 1988, Jaeger [8] posed the following conjecture which would imply both Berge–Fulkerson Conjecture and 5-CDCC.

Conjecture 1.1 The Petersen Coloring Conjecture [8]

Every bridgeless cubic graph has a Petersen-coloring.

The rest of the section is devoted to some alternative formulations of the Petersen coloring conjecture.

Let $G$ be a graph. A set of edges $C$ is a binary cycle if $C$ induces a subgraph of $G$ where every vertex has even degree. DeVos, Nešetřil and Raspaud [3] defined that, given two graphs $G$ and $H$, a mapping $\varphi :E\left(G\right)\to E\left(H\right)$ is cycle-continuous if the pre-image of each binary cycle of $H$ is a binary cycle of $G$. When both $G$ and $H$ are cubic and additionally $H$ is cyclically 4-edge-connected, $G$ has a cycle-continuous mapping to $H$ if and only if $G$ has an $H$-coloring. This leads to the first alternative formulation of the Petersen coloring conjecture.

Theorem 1.2 e.g. [1]

A cubic graph has a Petersen-coloring if and only if it has a cycle-continuous mapping to the Petersen graph.

However, the study on cycle-continuous mapping makes no progress on solving the Petersen coloring conjecture so far.

Consider Cremona–Richmond configuration $G_{cr}$, which has 15 points and 15 lines, as drawn in Fig. 1. A CR-coloring of a cubic graph $G$ is a mapping from $E\left(G\right)$ to the points of $G_{cr}$ such that any three mutually adjacent edges of $G$ are mapped to three points of $G_{cr}$ that lie in a line.

Theorem 1.3 [13]

A cubic graph has a Berge–Fulkerson cover if and only if it has a CR-coloring.

The truth of this theorem easily follows from a labeling of Cremona–Richmond configuration by $\{i,j\}$ with $1\le i<j\le 6$, as shown in Fig. 1. Here, we give another labeling of Cremona–Richmond configuration which yields that every CR-coloring of the graph $G$ is a nowhere-zero flow of $G$, that is, the flow values around a vertex sum up to zero. Such a labeling, depicted in Fig. 2, takes 15 non-zero elements of ${\mathbb{Z}}_2^4$.

Let $L_{cr}$ be a set of 10 lines obtained from the lines of $G_{cr}$ by removing 5 pairwise disjoint lines. The dotted lines in Fig. 2 indicate an example of $L_{cr}$.

Theorem 1.4 [13]

A cubic graph has a Petersen-coloring if and only if it has a CR-coloring using lines from $L_{cr}$.

From the previous two theorems, it is easy to see again that the Petersen coloring conjecture implies Berge–Fulkerson conjecture.

Unfortunately, the study on CR-colorings makes no progress on solving the Petersen coloring conjecture either. Here, we focus on another alternative formulation of the Petersen coloring conjecture, in terms of normal 5-edge-colorings.

Let $G$ be a cubic graph and $\varphi :E\left(G\right)\to \{1,2,\dots ,5\}$ be a proper 5-edge-coloring of $G$. An edge $e$ is poor (or rich) if $e$ together with its four adjacent edges uses precisely 3 (or 5) colors in total. An edge is normal if it is either rich or poor, and it is abnormal otherwise. A normal 5-edge-coloring is a proper 5-edge-coloring such that all the edges are normal. Jaeger [7] showed the following equivalence between Petersen colorings and normal 5-edge-colorings for cubic graphs.

Theorem 1.5 [7]

A cubic graph has a Petersen-coloring if and only if it has a normal 5-edge-coloring.

A possible minimal counterexample to the Petersen coloring conjecture was characterized in the literature. Jaeger [8] proved that it must be a cyclically 4-edge-connected snark. By the study on normal 5-edge-colorings of cubic graphs, Hägglund and Steffen [6] showed that the minimal counterexample does not contain $K_{3,3}^{\ast }$ as a subgraph (see Fig. 3 for $K_{3,3}^{\ast }$).

A quite few classes of cubic graphs were confirmed to have a normal 5-edge-coloring and thus a Petersen coloring as well. In [6] it showed that a cubic graph $G$ has a normal 5-edge-coloring when $G$ is a flower snark or a Goldberg snark or a generalized Blanuša snark of type 1 or 2. With the aid of computer, Brinkmann et al. [2] tested the Petersen coloring conjecture on cubic graphs of small order, and showed that every cubic graph of order no more than 36 has a normal 5-edge-coloring. Recently, Ferrarini, Mazzuoccolo and Mkrtchyan [4] confirmed the existence of normal 5-edge-colorings for a family of Loupekhine snarks.

Let $G$ be a cubic graph and $S_3$ a list of three 1-factors $M_1,M_2,M_3$ of $G$. For $0\le i\le 3$, let $E_i$ be the set of edges that are contained in precisely $i$ elements of $S_3$. Let $\left|E_0\right|=k$. The $k$-core of $G$ with respect to $S_3$ (or to $M_1,M_2,M_3$) is the subgraph $G_c$ of $G$ which is induced by $E_0\cup E_2\cup E_3$; that is, $G_c=G[E_0\cup E_2\cup E_3]$. If the value of $k$ is irrelevant, then we say that $G_c$ is a core of $G$. Define ${\mu }_3\left(G\right)=min\{k:G\text{has a}k\text{-core}\}$. Clearly, every bridgeless cubic graph has a ${\mu }_3\left(G\right)$-core. For any core, $E_0\cup E_2$ induces disjoint circuits.

Cores were introduced by Steffen [17] recently and used to prove partial results on some hard conjectures which are related to 1-factors of cubic graphs, such as Berge conjecture, Fan–Raspaud conjecture and some conjectures on cycle covers. The parameter ${\mu }_3\left(G\right)$ can measure how far a cubic graph $G$ is from being 3-edge-colorable, and it was related to some other parameters, such as girth and oddness. We refer to [5] for a survey on these kinds of measurements, and [9], [10], [11] for studies on cores and ${\mu }_3$. In this paper, we will use them to prove a partial result on the Petersen coloring conjecture.

Considering that a normal 5-edge-coloring requires each edge to be normal, Šámal [18] presented a weaker problem approximate to the Petersen coloring conjecture, that is, to search for a proper 5-edge-coloring such that the normal edges are as many as possible. Here, such a coloring is called a partially normal 5-edge-coloring. Later on, Bílková proved that a generalized prism has a proper 5-edge-coloring with two thirds of the edges normal ([1], Theorem 2.3) and that a cubic graph of large girth has a proper 5-edge-coloring with almost half of the edges normal ([1], Theorem 3.6). In this paper, we show that for every bridgeless cubic graph, there exists a proper 5-edge-coloring such that almost all the edges are normal. More precisely, we prove the following theorem.

Theorem 1.6

Every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $\left|E\left(G\right)\right|-{\mu }_3\left(G\right)$ many edges are normal.

Kaiser, Král and Norine [12] showed that any bridgeless cubic graph with $m$ edges contains three perfect matchings whose union covers at least $\frac{27}{35}m$ edges. That is to say, ${\mu }_3\left(G\right)\le \frac{8}{35}\left|E\left(G\right)\right|$ by the definition of ${\mu }_3$. So, a direct consequence of Theorem 1.6 is as follows.

Corollary 1.7

Every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $\frac{27}{35}\left|E\left(G\right)\right|$ edges are normal.

Normal or partially normal $k$-edge-colorings of cubic graphs were studied in several papers. Mazzuoccolo and Mkrtchyan [15] proved that every simple cubic graph $G$ (not necessarily bridgeless) admits a normal 7-edge-coloring, where the number 7 cannot be lowered down. They [14] also proved that any claw-free bridgeless cubic graph, permutation snark or tree-like snark admits a normal 6-edge-coloring and that any bridgeless cubic graph $G$ admits a 6-edge-coloring such that at least $\frac{7}{9}\left|E\left(G\right)\right|$ edges of $G$ are normal. Moreover, Pirot, Sereni and Škrekovski [16] showed that every bridgeless cubic graph $G$ admits a 4-edge-coloring such that at least $\frac{7}{15}\left|E\left(G\right)\right|$ edges of $G$ are normal, where the fraction $\frac{7}{15}$ is tight for Petersen graph.