We resolve the two-colour three-component conjecture of Fern\'andez, Pavez-Sign\'e and Stein for random bipartite graphs. More precisely, we prove that if $G\sim G(n,n,p)$ and $p\gg\sqrt{\log n/n}$, then with high probability every red--blue edge-colouring of $G$ admits a cover of its vertex set by at most three monochromatic connected components. The proof is based on a uniform expansion lemma for unions of common neighbourhoods and an alternating common-neighbourhood expansion argument.
We extend the conjecture of Erd\H{o}s and Gy\'arf\'as on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, th...
We prove that each $2$-edge-coloured spanning subgraph $G$ of $K_{n,n}$ with $\delta(G)\ge \lfloor (2n+1)/3 \rfloor$ can be covered by at most three monochromatic components. We provide a $2$-edge-coloured spanning subgraph of $K_{n,n}$ showing this minimum degree condition is sharp.
César Bispo, George Kontogeorgiou, M. Lage et al.· 0 citations
Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [...
Let $G$ be a Borel digraph with maximum out-degree $d \in \mathbb{N}$. We show that $G$ admits a random Borel $(2d+1)$-coloring for which every edge is almost surely not monochromatic. This gives a simpler proof of a recent result of Pelayo-G\'omez: such a graph $G$ admits a measurable proper $(2d+1)$-coloring with res...
Anton Bernshteyn, Edward Hou, Forte Shinko et al.· 0 citations
For a graph $H$, its $k$-colour vertex Ramsey class is the set of all graphs $G$ such that any colouring of the vertices of $G$ in $k$ colours results in a monochromatic (induced) copy of $H$. We prove that for any $k$, Ramsey classes of any non-isomorphic graphs are distinct.
Let $G$ be the line graph of a finite simple graph, with $n\geq1$ vertices and maximum degree $\Delta$. We prove that single-site Glauber dynamics for uniform proper $q$-colorings mixes in $O_\Delta(n\log(n/\varepsilon))$ steps for every integer $q\geq\Delta+5$. Our proof uses the Bochner framework of Chen and Liu (202...
Alireza Haqi· 0 citations
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