Snarks with special spanning trees

Cycle double covers and non-separating cycles

“…Recall that a decomposition of a graph G is a set of edge-disjoint subgraphs covering E(G). Hence, if a connected cubic graph G has a decomposition into a tree T and a cycle C, then C is a non-separating cycle of G. Note that all snarks with less than 38 vertices have a decomposition into a tree and a cycle and that there are infinitely many snarks with such a decomposition, see [12]. We consider the following reformulation (see Proposition 1.4) of the above conjecture.…”

“…Conversely, every cubic graph with a hist has trivially a decomposition into a tree and a cycle. For informations and examples of snarks with hists, see [12,13]. Let us also mention that Conjecture 1.3 limited to snarks is stated in [12].…”

“…For informations and examples of snarks with hists, see [12,13]. Let us also mention that Conjecture 1.3 limited to snarks is stated in [12].…”

Special Hist-Snarks

“…17 (the Hist is illustrated in bold face). For more informations on Hist-snarks, see [3]. The following definition is essential.…”

“…

A Hist in a cubic graph G is a spanning tree T which has only vertices of degree three and one. A snark with a Hist is called a Hist-snark, see [3]. We present several computer generated Hist-snarks which form generalizations of the Petersen graph.

…”