Related papers: The Lusternik-Schnirelmann theorem for graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but…

A set $D\subseteq V$ of a graph $G=(V,E)$ is called a restrained dominating set of $G$ if every vertex not in $D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textsc{Minimum Restrained Domination} problem is to…

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy…

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

We define the cover number of a graph $G$ by a graph class $\mathcal P$ as the minimum number of graphs of class $\mathcal P$ required to cover the edge set of $G$. Taking inspiration from a paper by Harary, Hsu and Miller, we find an exact…

The smallest number of cliques, covering all edges of a graph $ G $, is called the (edge) clique cover number of $ G $ and is denoted by $ cc(G) $. It is an easy observation that for every line graph $ G $ with $ n $ vertices, $cc(G)\leq n…

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and…

A graph $G$ is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$ where $\chi(G)$ denotes the chromatic number of $G$. We show that there are only finitely many $k$-critical $(P_3+\ell P_1)$-free graphs for all $k$ and…

The isoperimetric constant of a graph $G$ on $n$ vertices, $i(G)$, is the minimum of $\frac{|\partial S|}{|S|}$, taken over all nonempty subsets $S\subset V(G)$ of size at most $n/2$, where $\partial S$ denotes the set of edges with…

The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <=…

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…

Let $G$ be a finite simple graph. An independent set $I$ of $G$ is critical if $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$ for every independent set $J$ of $G$. A critical independent set is maximum if it has…

The union vertex-distinguishing chromatic index $\chi'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges…

Given two sets $A$ and $B$ of integers, we consider the problem of finding a set $S \subseteq A$ of the smallest possible cardinality such the greatest common divisor of the elements of $S \cup B$ equals that of those of $A \cup B$. The…