Double covers of cubic graphs with oddness 4

Abstract

We prove that a cubic 2-connected graph which has a 2-factor containing exactly 4 odd cycles has a cycle double cover.

Avoiding 5-Circuits in 2-Factors of Cubic Graphs

“…Since every cubic graph has an even number of vertices, its oddness must be even. Oddness is an interesting property to consider since the 5-flow conjecture and the cycle double cover conjecture are proven for snarks of small oddness [11,7,5]. Other parameters quantifying the uncolourability of cubic graphs can be related to oddness.…”

Cycle double covers and spanning minors II

“…In [4,5] it was proven that a graph with a two-factor with at most four odd cycles has a CDC. Although it was not noted in there this implies that a graph with a cycle of length l, l ∈ {n, .…”

“…If we start with the CDC C given by the face cycles on a planar embedding of the cube we see that there are only two non-isomorphic ways to add edges to Q 3 not stringing a cycle in C: either with endpoints residing on (1,2) and (3,4) or endpoints residing on (1,2) and (4,5). In Figs.…”

“…29 we show a Kotzig colouring of the graph obtained. (d) Adding an edge with endpoints residing on edges (5,7) and (3,4). The edge added here brings us back to case (1).…”

Cycle double covers and spanning minors I

“…In [12] Huck and Kochol proved that a cubic bridgeless graph with o(G) = 2 has a 5-CDC and more recently Häggkvist and McGuinness [8] and independently Huck [11] proved that a cubic graph with o(G) = 4 has a CDC. As a corollary of the result on oddness 2 a cubic bridgeless graph with a hamiltonian path has a 5-CDC, thus improving Tarsi's seminal theorem in [26], that every cubic bridgeless graph with a hamiltonian path has a CDC (Tarsi obtained a 6-CDC).…”