Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length l for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph on n > 4k^4 vertices with independence number k, then G is pancyclic. He then suggested that n = \Omega(k^2) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > ck^{7/3} suffices

Generating random graphs in biased Maker-Breaker games

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=\Theta\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all spanning trees having maximum degree $\Delta=O(1)$ with a bare path of linear length (a bare path in a tree $T$ is a path with all interior vertices of degree exactly two in $T$)

Pancyclicity of Hamiltonian and highly connected graphs

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $\kappa(G)$ for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a disconnected graph. A celebrated theorem of Chv\'atal and Erd\H{o}s says that $G$ is Hamiltonian if $\kappa(G) \ge \alpha(G)$. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if $\kappa(G) \ge 600\alpha(G)$ then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree $\delta(G) \ge 600\alpha(G)$ then G is pancyclic. Improving an old result of Erd\H{o}s, we also show that G is pancyclic if it is Hamiltonian and $n \ge 150\alpha(G)^3$. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if $\delta(G) \ge 300\alpha(G)$ then G contains a cycle of length $\ell$ for all $3 \le \ell \le \delta(G)/81$.Comment: 15 pages, 1 figur

Hamiltonian degree sequences in digraphs

We show that for each \eta>0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^- _1 \le ... \le d^-_n satisfy (i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i and (ii) d^-_i \geq i+ \eta n or d^+_{n-i- \eta n} \geq n-i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chv\'atal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.Comment: 17 pages, 2 figures. Section added which includes a proof of a conjecture of Thomassen for large tournaments. To appear in JCT