Related papers: Greedoids from flames

Graph homomorphism has been an important research topic since its introduction [17]. Stated in the language of binary relational structures in that paper [17], Lov\'asz proved a fundamental theorem that, for a graph $H$ given by its $0$-$1$…

We first give an alternative proof of the Alon-Tarsi list coloring theorem. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon-Tarsi Theorem: Let $G$ be a graph and let $D$ be…

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erd\H os--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov\'asz and Simonovits from…

A classical result from graph theory is that every graph with chromatic number \chi > t contains a subgraph with all degrees at least t, and therefore contains a copy of every t-edge tree. Bohman, Frieze, and Mubayi recently posed this…

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…

We consider the curve graph in the cases where it is not a Farey graph, and show that its Gromov boundary is linearly connected. For a fixed center point c and radius r, we define the sphere of radius r to be the induced subgraph on the set…

We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Dragani\'c, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently…

We consider the directed Min-Cost Rooted Subset $k$-Edge-Connection problem: given a digraph $G=(V,E)$ with edge costs, a set $T \subseteq V$ of terminals, a root node $r$, and an integer $k$, find a min-cost subgraph of $G$ that contains…

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented…

A well-known result of Nosal states that a graph $G$ with $m$ edges and $\lambda(G) > \sqrt{m}$ contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if $\lambda (G) >…

In 1968, Erd\"os and Lov\'asz conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=\chi(G) > \omega(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $\chi(G|S)\geq s$ and $\chi(G|T)\geq…

We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as…

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every graph is common.…

The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$,…

A keyring is a graph obtained by appending $r \geq 1$ leaves to one of the vertices of a cycle. We prove that for every $r \leq (k-1)/2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$…

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*}…

Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gr\"obner bases and initial ideals, it will be proved that, given integers $f$ and $d$ with $7 \leq f \leq d$, there exists a finite graph $G$ on…

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…

Given a multigraph $G$ and function $f : V(G) \rightarrow \mathbb{Z}_{\ge 2}$ on its vertices, a degree-$f$ subgraph of $G$ is a spanning subgraph in which every vertex $v$ has degree at most $f(v)$. The degree-$f$ arboricity $a_f(G)$ of…