Snarks from a Kászonyi perspective: A survey

Abstract

This is a survey or exposition of a particular collection of results and open problems involving snarks -simple "cubic" (3-valent) graphs for which, for nontrivial reasons, the edges cannot be 3-colored. The results and problems here are rooted in a series of papers by László Kászonyi that were published in the early 1970s. The problems posed in this survey paper can be tackled without too much specialized mathematical preparation, and in particular seem well suited for interested undergraduate mathematics students to pursue as independent research projects. This survey paper is intended to facilitate research on these problems.

Cubic Graphs with Colouring Defect 3

“…Kirchhoff's law yields that χ(r 1 ) + χ(r 2 ) + χ(r 3 ) = 0. By Proposition 3.1, χ is a nowherezero Z 3 2 -flow, therefore the values χ(r 1 ), χ(r 2 ), and χ(r 3 ) constitute a line ℓ in the Fano plane. Since C intersects R, the line {(0, 1, 1), (1, 0, 1), (1, 1, 0)} = {1, 2, 3} is excluded.…”

“…Since the complement of each M i in G is a 2-factor, it is easy to see that χ is a Z 3 2 -flow. We call χ the characteristic flow for M. Again, χ is a nowhere-zero Z 3 2 -flow if and only if G contains no triply covered edge. In the context of 3-arrays the characteristic flow was introduced in [14, p. 166], but the idea is older, see [27,Section 4].…”

“…If a 3-array M has no triply covered edge, the flow values of χ can be regarded as points of the Fano plane P G(2, 2) represented by the standard projective coordinates from Z 3 2 − {0}. In this representation, the points of P G(2, 2) are the non-zero elements of Z 3 2 and the lines of P G(2, 2) are the 3-element subsets {x, y, z} of Z 3 2 − {0} such that x + y + z = 0. The definition of the characteristic flow implies that for every vertex v of G the three flow values meeting at v form a line of the Fano plane.…”

“…On the left-hand side of the figure the points are labelled with the corresponding colours -subsets of {1, 2, 3}. Based on this correspondence we will refer to the colouring ϕ and the flow χ as the Fano colouring and the Fano flow associated with a 3-array M. The colours from {∅, 1, 2, 3, 12, 13, 23} and the corresponding elements of Z 3 2 − {0} will be used interchangeably. 2.…”

“…Our second theorem specifies conditions that must be satisfied by a snark K and a vertex v in order for the inflation of v to decrease the defect of K to 3. The formulation features an important concept of a cluster of 5-cycles in a snark that derives from relatively little known results of Kászonyi [20]- [22] and Bradley [1]- [3] concerning the structure of 3-edge-colourings of graphs. A cluster of 5-cycles in a cubic graph G, or simply a 5-cluster of G, is an inclusion-wise maximal connected subgraph of G formed by a union of 5-cycles.…”