On disjoint matchings in cubic graphs

In the paper graphs are assumed to be finite, undirected and without loops, though they may contain multiple edges. We will also consider pseudo-graphs, which, in contrast with graphs, may contain loops. Thus graphs are pseudo-graphs. We accept the convention that a loop contributes to the degree of a vertex by two. The set of vertices and edges of a pseudo-graph G will be denoted by V (G) and E(G), respectively. We also define: n = |V (G)| and m = |E(G)|. We will also follow to the scheme inherited from [ 20]: if G is a pseudo-graph and f is a graph-theoretic parameter, we will write just f instead of f(G). So, for example, if we would like to deal with the edge-set of a pseudo-graph G (0)∗ i , we will write E (0)∗ i instead of E(G (0)∗ i ); moreover we will write m (0)∗ i for the number of edges in this graph. A connected 2-regular graph with at least two vertices will be called a cycle. Thus, a loop is not considered to be a cycle in a pseudo-graph. Note that our notion of cycle differs from the cycles that people working on nowhere-zero flows and cycle double covers are used to deal with. The length of a path or a cycle is the number of edges lying on it. The path or cycle is even (odd) if its length is even (odd). Thus, an isolated vertex is a path of length zero, and it is an even path. For a graph G let ∆ = ∆(G) and δ = δ(G) denote the maximum and minimum degree of vertices in G, respectively. Let χ = χ(G) denote the chromatic class of the graph G.