1‐Factor and Cycle Covers of Cubic Graphs

Abstract

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E i be the set of edges contained in precisely i members of the k 1-factors. Let µ k (G) be the smallest |E 0 | over all lists of k 1-factors of G.Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Bergecovers and for the existence of three 1-factors with empty intersection. Furthermore, if µ 3 (G) = 0, then 2µ 3 (G) is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs.Cubic graphs with µ 4 (G) = 0 have a 4-cycle cover of length 4 3 |E(G)| and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang [18]. We also give a negative answer to a problem stated in [18].

Petersen Cores and the Oddness of Cubic Graphs

“…This paper focuses on the structure of cores and relates µ 2 , µ 3 to each other and to ω. In [14] it is shown that the girth of a snark G is at most 2µ 3 (G). We show that ω(G) ≤ µ 3 (G), then every µ 3 (G)-core of G is a very specific cyclic core.…”

On Hypohamiltonian and Almost Hypohamiltonian Graphs

“…[14,94,116,122]. Hypohamiltonian snarks have been studied in connection with the famous Cycle Double Cover Conjecture, see [14] for more details, and Sabidussi's Compatibility Conjecture [40].…”

“…Therefore, every cubic hypohamiltonian graph with chromatic index 4 must be a snark. In 2015, Steffen [122] published a conjecture on hypohamiltonian snarks-see Problem 3 in Chapter 6. In the recent not yet pub- 3 The chromatic index of a graph is the smallest number of colours necessary to colour the edges of the graph such that any two edges sharing an end-point do not have the same colour.…”

“…For this paragraph, we follow recent work of Steffen [122]. He writes that Jaeger and Swart [75] conjectured that (i) the girth and (ii) the cyclic connectivity of a snark is at most 6.…”

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

“…It turns out that the Cycle Double Cover Conjecture also has interesting connections with the excessive index of snarks. Indeed, it was proved independently by Steffen and Hou et al. that it is enough to prove the Cycle Double Cover Conjecture for snarks with excessive index at least 5.…”