Graph whose edges are in small cycles

Abstract

Paulraja (1987) conjectured the following:

• (i) If every edge of a 2-connected graph G lies in a cycle of length at most 4 in G, then G has a dominating closed trail.

• (ii) If, in addition, δ(G)⩾3, then G has a closed spanning trail.

  Collapsible graphs are defined and studied by Catlin (1988). Catlin showed that if H is a collapsable subgraph of G, then G has a spanning closed trail if and only if G/H, the graph obtained from G by contracting H, has a spanning closed trail. Catlin (1987) conjectured that a graph satisfying the hypothesis of (ii) is collapsable. In this paper, all three conjectures are proved.