Related papers: Polarity on $H$-split graphs

In this paper we study a multi-partite version of the Erd\H{o}s--Stone theorem. Given integers $r<k$ and $t\ge 1$, let $\text{ex}_k(n, K_{r+1}(t))$ be the maximum number of edges of $K_{r+1}(t)$-free $k$-partite graphs with $n$ vertices in…

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…

Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be…

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a…

The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of…

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…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on…

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$…

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…

An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is…

We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we…

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $\deg(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in…

The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

Given a graph $G = (V,E)$, a set $S \subset V$ is called a $k$-\emph{metric generator} for $G$ if any pair of different vertices of $G$ is distinguished by at least $k$ elements of $S$. A graph is $k$-\emph{metric dimensional} if $k$ is the…

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every…

We study the problem of transforming bipartite graphs into bicluster graphs. Abu-Khzam, Isenmann, and Merchad [IWOCA '25] introduced two variants of this problem. In both problems, the goal is to transform a bipartite graph into a bicluster…

The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no…