For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K1,4-free graph G with minimum degree δ(G) ≥ k+ 3 is pancyclic mod k and every claw-free graph G with δ(G) ≥ k+ 1 is pancyclic mod k, which confirms Thomassen’s conjecture [8] for claw-free graphs.

Some known results on claw-free (K 1;3-free) graphs are generalized to the larger class of almost claw-free graphs which were introduced by Ryjj a cek. In particular, we show that a 2-connected almost claw-free graph is 1-tough, and that a 2-connected almost claw-free graph on n vertices is hamiltonian if 1 3 (n ? 2), thereby (partly) generalizing results of Matthews and Sumner. Finally, we use...

For each proper subgraph H of K5, we determine all pairs (k, d) such that every H-minor-free graph is (k, d)-choosable or (k, d)-choosable. The main structural lemma is that the only 3-connected (K5 − e)-minor-free graphs are wheels, the triangular prism, and K3,3; this is used to prove that every (K5 − e)-minor-free graph is 4-choosable and (3, 1)-choosable.

For s ≥ 3 a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Cycles in K1,3-free graphs, called clawfree graphs, have beenwell studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K1,s-free graphs, normally called generalized claw-free graphs. In particular, we prove that if G is K1,s-fre...

The ratio of the connected domination number, γc, and the domination number, γ, is strictly bounded from above by 3. It was shown by Zverovich that for every connected (P5, C5)-free graph, γc = γ. In this paper, we investigate the interdependence of γ and γc in the class of (Pk, Ck)-free graphs, for k ≥ 6. We prove that for every connected (P6, C6)-free graph, γc ≤ γ+1 holds, and there is a fam...

The bull is a graph consisting of a triangle and two pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is a summary of the last two papers [2, 3] in a series [1, 2, 3]. The goal of the series is to give a complete description of all bull-free graphs. We call a bull-free graph elementary if it does not contain an induced three-edge-path P such that some vert...

We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical (P7, Ck)-free graphs, for both k = 4 and k = 5. We also show that there are only finitely many 4-critical (P8, C4)-free graphs. For each case of these cases we also give the complete lists of critical graphs and verte...

We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical (P7, Ck)-free graphs, for both k = 4 and k = 5. We also show that there are only finitely many 4-critical graphs (P8, C4)free graphs. For each case of these cases we also give the complete lists of critical graphs and...