Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by {\L}uczak, R\"odl, and Szemer\'edi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomass\'e in 2010. In this paper, we co...
Pedro Araújo, Xiao-Chuan Liu, Taísa L. Martins et al.· 0 citations
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 monoch...
We study the size of the largest monochromatic connected component that must appear in any edge-coloring of a random graph. Let $G\sim G(n,p)$ with $p\gg 1/n$ and $p=o(1)$, and write $np=he^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at l...
A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all...
Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\chi'_k(G)$ denote the minimum number of colors required, and let $\chi'_k$ be the supremum of $\chi'_k(G)$ over all finite simple graphs $G$....
Given a graph $H$, the minimum feedback vertex number of $H$ is the minimum number of vertices whose removal results in an acyclic graph. In this paper, we investigate Tur\'an-type extremal problems for bipartite graphs in terms of their feedback vertex number. Our first result concerns bipartite graphs $H$ with minimu...
Xiao-Chuan Liu, Xu Yang· 0 citations
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