Related papers: Burling graphs as intersection graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of…

Given a non-decreasing sequence $S=(s\_1,s\_2, \ldots, s\_k)$ of positive integers, an {\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\{s\_1,s\_2, \ldots, s\_k\}$ such that any two vertices with color $s\_i$ are…

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $\chi_S(G)$. In this paper, $\chi_S$-critical graphs are introduced as the graphs $G$ such that…

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

Given a graph $G$, a vertex switch of $v \in V(G)$ results in a new graph where neighbors of $v$ become nonneighbors and vice versa. This operation gives rise to an equivalence relation over the set of labeled digraphs on $n$ vertices. The…

A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all…

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…

A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has…

We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their…

The chromatic discrepancy of a graph $G$, denoted $\phi(G)$, is the least over all proper colourings $\sigma$ of $G$ of the greatest difference between the number of colours $|\sigma(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its…

A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection…

In 1968, Erd\"os defined the Shift Graph as the graph whose vertices are the $k$-element subsets of $[n]=\{0,1,2,...,n-1\}$ such that $A=\{a_1,...,a_k\}$ and $B=\{b_1,...,b_k\}$ are neighbours iff $a_1<b_1=a_2<b_2=a_3<... <b_{n-1}=a_n<b_n$.…

We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all…

A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

We provide a finite forbidden induced subgraph characterization for the graph class $\varUpsilon_k$, for all $k \in \mathbb{N}_0$, which is defined as follows. A graph is in $\varUpsilon_k$ if for any induced subgraph, $\Delta \leq \chi -1…

For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors…