Related papers: Short monochromatic odd cycles

In 1964 Erd\H{o}s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges.…

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a…

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon,…

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common…

Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, and Szemer\'edi proved Lehel's conjecture for large $n$, Allen gave a different proof…

In 1959, Erd\H{o}s and Gallai proved that every graph G with average vertex degree ad(G)\geq 2 contains a cycle of length at least ad(G). We provide an algorithm that for k\geq 0 in time 2^{O(k)} n^{O(1)} decides whether a 2-connected…

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We…

In strengthening a result of Andr\'asfai, Erd\H{o}s and S\'os in 1974, H\"{a}ggkvist proved that if $G$ is an $n$-vertex $C_{2k+1}$-free graph with minimum degree $\delta(G)>\frac{2n}{2k+3}$ and $n>\binom{k+2}{2}(2k+3)(3k+2)$, then $G$…

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the…

A 2018 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \bmod k$ as the complete graph on $k+1$ vertices.…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

An odd coloring of a graph is a proper coloring such that every non-isolated vertex has a color that appears at an odd number of its neighbors. This notion was introduced by Petr\v{s}evski and \v{S}krekovski in 2022. In this paper, we focus…

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given…

We prove that in every $2$-edge-colouring of $K_n$ there is a collection of $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, thus confirming a conjecture of Erd\H{o}s. We also prove a corresponding stability result, showing that…

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum…