arXiv subjects

Andrew Treglown

Publications and source records attributed to Andrew Treglown.

60 records · Page 4Linked to original sources

arXiv2012

Embedding spanning bipartite graphs of small bandwidth

Boettcher, Schacht and Taraz gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobas and Komlos. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

math.CO↗

arXiv2012

Monochromatic triangles in three-coloured graphs

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.

math.CO↗

arXiv2010

A note on some embedding problems for oriented graphs

We conjecture that every oriented graph $G$ on $n$ vertices with $δ^+ (G) , δ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered.

math.CO↗

arXiv2009

Hamilton decompositions of regular tournaments

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each η>0 every regular tournament G of sufficiently large order n contains at least (1/2-η)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.

math.CO↗

arXiv2009

Hamiltonian degree sequences in digraphs

We show that for each η>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+ ηn or d^-_{n-i- ηn} \geq n-i and (ii) d^-_i \geq i+ ηn or d^+_{n-i- ηn} \geq n-i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

math.CO↗

arXiv2009

An Ore-type theorem for perfect packings in graphs

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let δ_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} δ_{\rm Ore} (H,n)/n.

math.CO↗