Related papers: K_4-free graphs have sparse halves
In this paper, we present a sharp upper bound for the spectral radius of an $n$-vertex graph without $F$-minor for sufficient large $n$, where $F$ is obtained from the complete graph $K_r$ by deleting disjointed paths. Furthermore, the…
We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the…
A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…
By considering graphs as discrete analogues of Riemann surfaces, Baker and Norine (Adv. Math. 2007) developed a concept of linear systems of divisors for graphs. Building on this idea, a concept of gonality for graphs has been defined and…
A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the…
Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…
A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…
In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices,…
We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge…
A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erd\H{o}s, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is,…
A drawing of a graph is $k$-plane if no edge is crossed more than $k$ times. In this paper we study saturated $k$-plane drawings with few edges. This are $k$-plane drawings in which no edge can be added without violating $k$-planarity. For…
Let $G$ be a graph and $S$ be a set of cliques of $G$. The set $S$ is an indeque set if every component of $G[S]$, the subgraph induced by vertices of $S$, is a clique. In this paper, we prove that the indeque ratio of $K_4$-minor-free…
We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $\alpha(T)-1$ parts certifying this fact. Each part induces a graph which is…
It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.
An ISK4 in a graph G is an induced subgraph of G that is isomorphic to a subdivision of K4 (the complete graph on four vertices). A wheel is a graph that consists of a chordless cycle, together with a vertex that has at least three…
A fork is a graph obtained from $K_{1,3}$ (usually called claw) by subdividing an edge once. A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and…
We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected,…
A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal…
A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Despite the existence of infinite families of…