91 research outputs found
Forcing spanning subgraphs via Ore type conditions
Abstract We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all â, Îł > 0 there are ÎČ, n 0 > 0 such that for all n â„ n 0 the following holds. Let G = (V, E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree â(H) †â and bandwidth bw(H) †ÎČn, and G satisfies deg(u) + deg(v) â„
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders
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.Comment: 23 pages, file updated, to appear in Combinatorics, Probability and
Computin
Recommended from our members
Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1â7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, ïŹnite or countable structures â discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
On degree sequences forcing the square of a Hamilton cycle
A famous conjecture of P\'osa from 1962 asserts that every graph on
vertices and with minimum degree at least contains the square of a
Hamilton cycle. The conjecture was proven for large graphs in 1996 by Koml\'os,
S\'ark\"ozy and Szemer\'edi. In this paper we prove a degree sequence version
of P\'osa's conjecture: Given any , every graph of sufficiently
large order contains the square of a Hamilton cycle if its degree sequence
satisfies for all . The degree sequence condition here is asymptotically best possible. Our
approach uses a hybrid of the Regularity-Blow-up method and the
Connecting-Absorbing method.Comment: 52 pages, 5 figures, to appear in SIAM J. Discrete Mat
Robust expansion and hamiltonicity
This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ârobustly expandingâ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in KĂŒhn and Osthusâs proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments.
In Chapters 3 and 4, we prove that every sufficiently large 3-connected -regular graph on vertices with â„ n/4 contains a Hamilton cycle. This answers a problem of BollobĂĄs and HĂ€ggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this.
Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the largest degree in a sufficiently large graph on n vertices is at least a little larger than /3 + for †/3, then contains the square of a Hamilton cycle
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