A note on the multicolor size-Ramsey numbers of connected graphs

[2404.05856] A note on the multicolor size-Ramsey numbers of connected graphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2404.05856 (math) [Submitted on 8 Apr 2024 ( v1 ), last revised 8 Apr 2026 (this version, v3)] Title: A note on the multicolor size-Ramsey numbers of connected graphs Authors: Louis DeBiasio View a PDF of the paper titled A note on the multicolor size-Ramsey numbers of connected graphs, by Louis DeBiasio View PDF HTML (experimental) Abstract: The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich proved that $\widehat{R}_r(P_{m+1})=\Omega(r^2m)$ where $P_{m+1}$ is the path on $m$ edges. He explains that his proof actually applies to any connected graph $H$ with $m$ edges and vertex cover number larger than $\sqrt{m}$. He also notes that some restriction on the vertex cover number is necessary since the star with $m$ edges, $K_{1,m}$, has vertex cover number 1 and satisfies $\widehat{R}_r(K_{1,m})=r(m-1)+1$. We prove that the star is actually the only exception; that is, $\widehat{R}_r(H)=\Omega(r^2m)$ for every non-star connected graph $H$ with $m$ edges. We also prove a strengthening of this result for trees. It follows from results of Beck and Dellamonica that $\widehat{R}_2(T)=\Theta(\beta(T))$ for every tree $T$ with bipartition $\{V_1, V_2\}$ and $\beta(T)=|V_1|\max\{d(v):v\in V_1\}+|V_2|\max\{d(v):v\in V_2\}$. We prove that $\widehat{R}_r(T)=\Omega(r^2\beta(T))$ for every tree $T$, again with the exception of the star. Additionally, we prove that for the family of non-star trees $T$ with $\beta(T)=\Omega(n_1n_2)$ (which includes all non-star trees of linear maximum degree and all trees of radius 2 for example) we have $\widehat{R}_r(T)=\Theta(r^2\beta(T))$. Comments: (v3) 16 pages (plus a 2-page appendix), final revisions. In particular, fixed an error in the appendix which significantly changed the statement of Theorem 7.4; (v2) 15 pages (plus a 2-page appendix), many revisions throughout the paper in response to referee reports; (v1) 11 pages (plus a 1-page appendix) Subjects: Combinatorics (math.CO) Cite as: arXiv:2404.05856 [math.CO] (or arXiv:2404.05856v3 [math.CO] for this version) https://doi.org/10.48550/arXiv.2404.05856 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Louis DeBiasio [ view email ] [v1] Mon, 8 Apr 2024 20:40:50 UTC (12 KB) [v2] Wed, 6 Aug 2025 18:08:21 UTC (17 KB) [v3] Wed, 8 Apr 2026 20:58:49 UTC (19 KB) Full-text links: Access Paper: View a PDF of the paper titled A note on the multicolor size-Ramsey numbers of connected graphs, by Louis DeBiasio View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2024-04 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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