Related papers: The Erd\H{o}s-Faudree Problems and the Isolate-Fre…

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number,…

A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free…

In a recent paper, Gerbner, Patk\'{o}s, Tuza and Vizer studied regular $F$-saturated graphs. One of the essential questions is given $F$, for which $n$ does a regular $n$-vertex $F$-saturated graph exist. They proved that for all…

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169…

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge…

Given an $n$-vertex graph $G$, let $\hom (G)$ denote the size of a largest homogeneous set in $G$ and let $f(G)$ denote the maximal number of distinct degrees appearing in an induced subgraph of $G$. The relationship between these…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

For every natural number $d$, we construct finite $d$-regular simple graphs that, for every $r \le d$, contain an independent exact $r$-cover. This answers a question of Gray and Johnson that arose in their study of 2-step transit…

The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong…

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset…

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or…

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient…

In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$,…

We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the…

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have…

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs,…

A connected r-regular graph, where $r \geq 3$, is an r-graph if each odd cut has at least r edges. Every r-graph is matching covered - a connected graph whose each edge participates in some perfect matching. We set out to: (i) characterize…