As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a speciic innnite sequence of 3-colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that if a (not necessarily maximal) triangle-free graph of order n has minimum degree n=...

A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In a previous paper we gave a complete description of all 3-colourable prismatic graphs, and of a slightly more general class, the “orientable” prismatic graphs. In this paper we describe the non-orientable ones, thereby completing a description of all prismatic graphs. Since complements of pri...

A graph G is Nm-locally connected if for every vertex v in G, the vertices not equal to v and with distance at most m to v induce a connected subgraph in G. We show that both connectedN2-locally connected claw-free graph and 3-edge-connected N3-locally connected claw-free graph have connected even [2, 4]-factors, which settle a conjecture by Li in [6].

We show that every 3-connected claw-free graph which contains no induced copy of P11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P12 this result is, in a way, best possible. 1. Statement of the main result A graph G is {H1, H2, . . . Hk}-free if G contains no induced subgraphs isomorphic to any of the graphs Hi, i = 1, 2, . . . , k. A...