Powers of Hamiltonian cycles in multipartite graphs

DeBiasio, Louis; Martin, Ryan; Molla, Theodore

doi:10.48550/arxiv.2106.11223

Cited by 1 publication

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that (k −1)th powers of tight cycles on kn vertices admit equitable k-colourings, thus in particular Theorem 2.11 shows the existence of (k − 1)th powers of Hamilton cycles in G, in all cases. In the particular case of (k − 1)th powers of cycles, Theorem 2.11 was shown to be true very recently by DeBiasio, Martin and Molla [19], whose setting also allowed for suitably imbalanced partite graphs.…”

Section: Multipartite Graphsmentioning

confidence: 90%

On sufficient conditions for Hamiltonicity in dense graphs

Lang¹,

Sanhueza‐Matamala²

2021

Preprint

View full text Add to dashboard Cite

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, Pósatype degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.structure, which are used as a black boxes. The proofs of these results can be found in Section 5.2.1. Ore-type conditions. Motivated by the Hajnal-Szemerédi Theorem, Kierstead and Kostochka [40] investigated optimal Ore-type conditions which ensure the existence of clique factors and proved the following result. Theorem 2.3 (Clique factors under Ore-type conditions). For n divisible by k, let G be a graph on n vertices with deg(x) + deg(y) 2 k−1 k n − 1 for all xy / ∈ E(G). Then G contains a k-clique factor.Note that the result is tight, as witnessed, for instance, by slightly imbalanced complete kpartite graphs. As an extension of this, Kühn, Osthus and Treglown [51] showed an Ore-type result for general factors (not just cliques). For sufficiently large graphs, Châu [13] proved a generalisation of Ore's theorem for squares of Hamilton cycles, and also conjectured generalisations of this for all k 3. For k = 2, Knox and Treglown [41] were able to strengthen Ore's theorem to (β, ∆, 2)-Hamiltonian graphs. They also conjectured corresponding extensions for all k 3. This was confirmed for k = 3 by Böttcher and Müller [6]. The following result proves these conjectures in a strong sense for all k 2. Theorem 2.4 (Bandwidth theorem under Ore-type conditions). For k, ∆ ∈ N and µ > 0, there are z, β > 0 and n 0 ∈ N with the following property. Let G be a graph on n n 0 vertices with deg(x) + deg(y) 2 k−1 k n + µn for all xy / ∈ E(G). Then G is (z, β, ∆, k)-Hamiltonian. 2.2. Pósa-type conditions. Balogh, Kostochka and Treglown [3, 4] studied degree conditions that guarantee the existence of clique factors and powers of Hamilton cycles. Treglown [71] proved the following Pósa-type result for clique factors. Theorem 2.5 (Clique factors under Pósa-type conditions). For k ∈ N and µ > 0, there is an n 0 ∈ N with the following property. Let G be a graph with degree sequence d 1 , . . . , d n where n n 0 is divisible by k. Suppose that d i k−2 k n + i + µn for every i n/k. Then G contains a k-clique factor.

show abstract

Section: Multipartite Graphsmentioning

confidence: 90%