A P-decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P 3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an-regular graph to admit a P-decomposition, for odd.… (More)
We study the decomposition conjecture posed by Barát and Thomassen (2006), which states that, for each tree T , there exists a natural number k T such that, if G is a k T-edge-connected graph and |E(T)| divides |E(G)|, then G admits a partition of its edge set into classes each of which induces a copy of T. In a series of papers, starting in 2008, Thomassen… (More)
Let H be an orientation of a graph H. Alon and Yuster proposed the problem of determining or estimating D(n, m, H), the maximum number of H-free orientations a graph with n vertices and m edges may have. We consider the maximum number of H-free orientations of typical graphs G(n, m) with n vertices and m edges. Suppose H = C is the directed cycle of length… (More)
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for… (More)
Our main result tells us that mild density and pseudorandom conditions allow one to prove certain counting lemmas for a restricted class of subhypergraphs in a sparse setting. As an application, we present a variant of a universality result of Rödl for sparse, 3-uniform hypergraphs contained in strongly pseudorandom hypergraphs.