Related papers: Clique immersion in graphs without fixed bipartite…

Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_{\alpha}, V_{\beta}\}$. We prove that for every vertex $v \in V_{\beta}$…

B. Bollob\'{a}s and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

We prove that for all $r\geq2$ and c>0, every graph of order n with at least cn^{r} cliques of order r contains a complete r-partite graph with each part of size $\lfloor c^{r}\log n \rfloor.$ This result implies a concise form of the…

A graph K is multiplicative if a homomorphism from any product G x H to K implies a homomorphism from G or from H. Hedetniemi's conjecture states that all cliques are multiplicative. In an attempt to explore the boundaries of current…

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le \omega(G)^{3t-3+o(1)}$.…

Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that…

In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3-space $\mathbb{R}^3$. An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if it sends every edge to…

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We initiate a…

Let $\mathcal{F}$ be a family of fixed graphs and let $d$ be large enough. For every $d$-regular graph $G$, we study the existence of a spanning $\mathcal{F}$-free subgraph of $G$ with large minimum degree. This problem is well-understood…

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model…

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by…

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$…

Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. We prove that if $B$ and $R$ are two…

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well.…

We present an easy structure theorem for graphs which do not admit an immersion of the complete graph. The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call…

We study the chromatic number of graphs that exclude a clique as a strong odd immersion and have independence number two. Given a graph $G$ and $t\in\mathbb{Z}^+$, we prove that if $\alpha(G)\leq 2$ and $G$ has no strong odd…

The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph $H$ there exists a constant $c(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least…

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each…

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $\chi(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2\chi(G))$…