Global cycle properties in graphs with large minimum clustering coefficient

The clustering coefficient of a vertex in a graph is the proportion of neighbours of the vertex that are adjacent. The minimum clustering coefficient of a graph is the smallest clustering coefficient taken over all vertices. A complete structural characterization of those locally connected graphs, with minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully cycle extendable is given in terms of strongly induced subgraphs with given attachment sets. Moreover, it is shown that all locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly pancyclic, thereby proving Ryjacek's conjecture for this class of locally connected graphs.Comment: 16 pages, two figure

Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42 pages, 9 figures, 20 pages of appendice

A Survey of Best Monotone Degree Conditions for Graph Properties

We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvatal's well-known degree condition for hamiltonicity is best possible.Comment: 25 page

Best monotone degree conditions for binding number

AbstractWe give sufficient conditions on the vertex degrees of a graph G to guarantee that G has binding number at least b, for any given b>0. Our conditions are best possible in exactly the same way that Chvátal’s well-known degree condition to guarantee a graph is Hamiltonian is best possible