Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proper Z4 x Z2-colorings: structural characterization with application to some snarks

Jelena Sedlar, Riste Škrekovski

Abstract

A proper abelian coloring of a cubic graph G by a finite abelian group A is any proper edge-coloring of G by the non-zero elements of A such that the sum of the colors of the three edges incident to any vertex v of G equals zero. It is known that cyclic groups of order smaller than 10 do not color all bridgeless cubic graphs, and that all abelian groups of order at least 12 do. This leaves the question open for the four so called exceptional groups Z4 x Z2, Z3 x Z3, Z10 and Z11 for snarks. It is conjectured in literature that every cubic graph has a proper abelian coloring by each exceptional group and it is further known that the existence of a proper Z4 x Z2-coloring of G implies the existence of a proper coloring of G by all the remaining exceptional groups. In this paper, we give a characterization of a proper Z4 x Z2-coloring in terms of the existence of a matching M in a 2-factor F of G with particular properties. Moreover, in order to modify an arbitrary matching M so that it meets the requirements of the characterization, we first introduce an incidence structure of the cycles of F in relation to the cycles of G - M. Further, we provide a sufficient condition under which M can be modified into a desired matching in terms of particular properties of the introduced incidence structure. We conclude the paper by applying the results to some oddness two snarks, in particular to permutation snarks. We believe that the approach of this paper with some additional refinements extends to larger classes of snarks, if not to all in general.

Proper Z4 x Z2-colorings: structural characterization with application to some snarks

Abstract

A proper abelian coloring of a cubic graph G by a finite abelian group A is any proper edge-coloring of G by the non-zero elements of A such that the sum of the colors of the three edges incident to any vertex v of G equals zero. It is known that cyclic groups of order smaller than 10 do not color all bridgeless cubic graphs, and that all abelian groups of order at least 12 do. This leaves the question open for the four so called exceptional groups Z4 x Z2, Z3 x Z3, Z10 and Z11 for snarks. It is conjectured in literature that every cubic graph has a proper abelian coloring by each exceptional group and it is further known that the existence of a proper Z4 x Z2-coloring of G implies the existence of a proper coloring of G by all the remaining exceptional groups. In this paper, we give a characterization of a proper Z4 x Z2-coloring in terms of the existence of a matching M in a 2-factor F of G with particular properties. Moreover, in order to modify an arbitrary matching M so that it meets the requirements of the characterization, we first introduce an incidence structure of the cycles of F in relation to the cycles of G - M. Further, we provide a sufficient condition under which M can be modified into a desired matching in terms of particular properties of the introduced incidence structure. We conclude the paper by applying the results to some oddness two snarks, in particular to permutation snarks. We believe that the approach of this paper with some additional refinements extends to larger classes of snarks, if not to all in general.
Paper Structure (10 sections, 12 theorems, 3 equations, 13 figures)

This paper contains 10 sections, 12 theorems, 3 equations, 13 figures.

Table of Contents

  1. Introduction
  2. A characterization of proper $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$-colorings
  3. Applying the characterization and two obstacles
  4. Oddness two snarks and the choice of $M$ which avoids problems P1) and P2).
  5. Oddness two snarks and the choice of $M$ with the problem P1).
  6. Oddness two snarks and the choice of $M$ with the problem P2).
  7. A sufficient condition for the existence of a proper $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$-coloring
  8. Loop-cycle-incidence graph.
  9. Application to oddness two snarks
  10. Concluding remarks

Key Result

Theorem 2

If a cubic graph has a proper $A$-coloring for $A=\mathbb{Z}_{4}\times\mathbb{Z}_{2}$, then it has a proper $A$-coloring for every $A\in\{\mathbb{Z}_{3}\times\mathbb{Z}_{3},\mathbb{Z}_{10},\mathbb{Z}_{11}\}$.

Figures (13)

  • Figure 1: An oddness two snark $G$ on 40 vertices and a $2$-factor $F$ in $G$ with two odd and two even cycles, where different cycles of $F$ are shown in different colors.
  • Figure 2: For the snark $G$ and the $2$-factor $F$ in $G$ depicted in Figure \ref{['Fig01']}, this figure shows a maximum matching $M$ in $F$ which consists of dashed edges.
  • Figure 3: For the snark $G$ from Figure \ref{['Fig01']}, and the maximum matching $M$ in $F$ from Figure \ref{['Fig02']}, this figure shows the corresponding graph $H=G-M$. Notice that the main component of $H$ is a $\Theta$-graph.
  • Figure 4: An oddness two snark $G$ on 26 vertices and a $2$-factor $F$ in $G$ with four cycles, two of which are odd. The different cycles of $F$ are shown in different colors.
  • Figure 5: For the snark $G$ and $2$-factor $F$ in $G$ depicted in Figure \ref{['Fig04']}, this figure shows a maximum matching $M$ in $F$ which consists of dashed edges such that the graph $H=G-M$ shown in Figure \ref{['Fig06a']} does not have an $F$-matching.
  • ...and 8 more figures

Theorems & Definitions (15)

  • Conjecture 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Corollary 5
  • Proposition 6
  • Proposition 7
  • Corollary 8
  • Theorem 9
  • Lemma 10
  • ...and 5 more