Related papers: Abundance: Asymmetric Graph Removal Lemmas and Int…

In this paper, we consider the following two problems: (i) Deletion Blocker($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with…

Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking…

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in…

The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at…

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that…

We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields…

King, Lu, and Peng recently proved that for $\Delta\geq 4$, any $K_\Delta$-free graph with maximum degree $\Delta$ has fractional chromatic number at most $\Delta-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$.…

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $\lambda$ of $A(G)$ is denoted by $m_\lambda(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$…

We consider the robust chromatic number $\chi_1(G)$ of planar graphs $G$ and show that there exists an infinite family of planar graphs $G$ with $\chi_1(G) = 3$, thus solving a recent problem of Bacs\'{o}~et~al. (The robust chromatic number…

An $r$-graph is an $r$-regular graph with no odd cut of size less than $r$. A well-celebrated result due to Lov\'asz says that for such graphs the linear system $Ax = \textbf{1}$ has a solution in $\mathbb{Z}/2$, where $A$ is the $0,1$ edge…

The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We…

In this paper, we show that for any positive integer $m$ and $k\in [2]$, let $G$ be a $(2m+2k+2)$-connected graph and let $a_1,\ldots , a_m, s, t$ be any distinct vertices of $G$, there are $k$ internally disjoint $s$-$t$ paths $P_1,…

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan…

A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have…

It is proved that for every $\varepsilon>0$, there exists $K>0$ such that for every integer $t\ge2$, every graph with chromatic number at least $Kt$ contains a minor with $t$ vertices and edge density at least $1-\varepsilon$. Indeed,…

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest…

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log…

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…

We address the problem of finding sets of integers of a given size with a maximum number of pairs summing to powers of $2$. By fixing particular pairs, this problem reduces to finding a labeling of the vertices of a given graph with…