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 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 least $n-\Theta(ne^{-h})$. Moreover, we construct colorings showing that this bound is best possible up to constant factors. We extend this result to three colors: for $p\gg 1/n$ and $p=o(1)$, with high probability every $3$-edge-coloring of $G$ contains a monochromatic connected component of size at least $\frac{n}{2}-\Theta(1/p)$, and this estimate is again tight up to constant factors. In the bipartite setting $G\sim G(n,n,p)$, under the same assumptions on $p$, we prove an analogous statement: with high probability, every $2$-edge-coloring contains two monochromatic components whose union covers all but $\Theta(ne^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.
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 proper edge-colorings of $G$. We prove that, for every fixed $0<\alpha<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $\alpha n$ satisfies $\operatorname{rpc}(G,c)\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.
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 $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\chi'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $\chi'_k(G_{k,c})=k+c+1$. Consequently, $\chi'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $\Delta(G_m)=\chi'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $\Delta(G_k)=\chi'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.
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 minimum feedback vertex number one. Such graphs can be obtained from a forest by identifying a specified collection of leaves into a single vertex. For these graphs, we show that $\text{ex}(n, H)$ is upper bounded by $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the shortest cycle contained in $H$. In addition, we consider a family of bipartite graphs with minimum feedback vertex number three. Let $E_{k,t}$ be the graph obtained from the theta graph $\theta_{k,t}$ by joining a new vertex $x$ to one side of the bipartition and another vertex $y$ to the other. Let $E^+_{k,t}$ denote the graph obtained by adding the edge $xy$ to $E_{k,t}$. We prove that for any $k\geq 2$ and sufficiently large $t$, $\text{ex}(n, E^+_{k,t})= \Theta(n^{\frac{3k-1}{2k-1}}).$
Xiao-Chuan Liu, Xu Yang· 0 citations
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