Let G be a graph and F : V ( G ) → 2 N be a mapping. The graph G is said to be F‐ avoiding if there exists an orientation O of G such that d O + ( v ) ∉ F ( v ) for every v ∈ V ( G ), where d O + ( v ) denotes the out‐degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and ∣ F ( v ) ∣ ≤ d G ( v ) / 2 for every v ∈ V ( G ), then G is F‐avoiding. The bound ∣ F ( v ) ∣ ≤ d G ( v ) / 2 is best possible. For every graph G, we conjecture that if ∣ F ( v ) ∣ ≤ 1 2 ( d G ( v ) − 1 ) for every v ∈ V ( G ), then G is F‐avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.

The edge clique cover number ecc ( G ) of a graph G is the size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw‐free, then ecc ( G ) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in n for ecc ( G ) for such graphs, improving upon the bou nd of O ( n 4 ∕ 3 log 1 ∕ 3   n ) due to Javadi, Maleki, and Omoomi. More precisely we prove that ecc ( G ) is at most the minimum of n + δ ( G ) and 2 n − normalΩ ( n log   n ), where δ ( G ) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ecc ( G )) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.

Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.

Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a recent result of Abuieda et al is nearly best possible.