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We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different…

A graph is apex if it becomes planar after the deletion of one vertex. The family of apex graphs is closed under taking minors, so it is characterized by a finite set of forbidden minors. Determining the finite set of forbidden minors for…

Petru\v{s}evski and \v{S}krekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a…

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the…

Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and $p_2+p_3=i$. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are…

A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. Fabrici and G\"oring proved that six colors are…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

A central vertex of a graph is a vertex whose eccentricity equals the radius. The center of a graph is the set of all central vertices. The central ratio of a graph is the ratio of the cardinality of its center to its order. In 1982,…

We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\' edli, we prove that the third, fourth and fifth largest numbers of congruences…

An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. Gupta et al. (2012,\cite{Gupta}) showed that every $n$-vertex graph has at most $10^{\frac{n}{5}}\approx 1.5849^n$ maximal induced matchings,…

A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite $1$-planar graphs with prescribed numbers of vertices in partite sets. Bipartite…

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let…

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…

A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all…

The fractional matching number of a graph G, is the maximum size of a fractional matching of G. The following sharp lower bounds for a graph G of order n are proved, and all extremal graphs are characterized in this paper. (1)The sum of the…

The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that…

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number…

The planar Tu\'{a}n number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for…

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $\mathrm{ex}(n, H, F)$. Let $(\ell+1) \cdot F$ denote $\ell+1$ vertex disjoint copies of $F$. In this paper, we…