Given a positive integer s, the s‐colour size‐Ramsey number of a graph H is the smallest integer m such that there exists a graph G with m edges with the property that, in any colouring of E(G) with s colours, there is a monochromatic copy of H. We prove that, for any positive integers k and s, the s‐colour size‐Ramsey number of the kth power of any n‐vertex bounded degree tree is linear in n. As a corollary, we obtain that the s‐colour size‐Ramsey number of n‐vertex graphs with bounded treewidth and bounded degree is linear in n, which answers a question raised by Kamčev, Liebenau, Wood and Yepremyan.

We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n-vertex graphs G with minimum degree n for arbitrarily small > 0. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that = k k+1 suffices for large n. For smaller values of the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density > 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least > 0, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalizes recent results of Staden and Treglown. KEYWORDS absorption method, bandwidth theorem, powers of Hamiltonian cycles 1 INTRODUCTION We study sufficient conditions for the existence of spanning subgraphs in large finite graphs and begin the discussion with powers of Hamiltonian cycles. For k ∈ N the kth power of a given graph H is the graph H k on the same vertex set with xy being an edge in H k if x and y are distinct vertices of H that are connected in H by a path of at most k edges. For simplicity, we refer to a kth power of a path with at least k vertices as a k-path. Moreover, we refer to the ordered k-tuples of the first and last k vertices This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

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MAESAKA, G. S. Graphs and hypergraphs with high girth and high chromatic number. 2018. 91f. Thesis