Related papers: Guessing Games on Triangle-free Graphs

We consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by…

The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph $G$ by two players, Alice…

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$…

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f…

Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned…

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that together cover all the vertices in $G$. In this paper we investigate perfect $H$-tilings in a random graph model introduced by Bohman,…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. We continue a recent study into the clique-width of $(H_1,H_2)$-free graphs and present three new classes of…

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge $\{u,w\}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of $H$-free split graphs whose clique-width is…

Let $\mathcal{H}$ be a 3-graph on $n$ vertices. The matching number $\nu(\mathcal{H})$ is defined as the maximum number of disjoint edges in $\mathcal{H}$. The generalized triangle $F_5$ is a 3-graph on the vertex set $\{a,b,c,d,e\}$ with…

We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem in $K_r$-minor free graphs, when there is an assignment satisfying $1-\varepsilon$…

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to…

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the…

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least \[ e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)} \] independent sets. This improves a recent result of the…

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.…

We study the hat guessing game on graphs. In this game, a player is placed on each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible colors. Each player makes a deterministic guess on their hat color based on the…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandst\"adt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandst\"adt, Le and Mosca…

A \emph{biclique} is a maximal bipartite complete induced subgraph of $G$. Bicliques have been studied in the last years motivated by the large number of applications. In particular, enumeration of the maximal bicliques has been of interest…