Related papers: The Extremal Function and Colin de Verdi\`{e}re Gr…

The Erd\H{o}s--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erd\H{o}s--Gallai Theorem was given by Kopylov: If $G$…

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of…

We study mechanisms that select a subset of the vertex set of a directed graph in order to maximize the minimum indegree of any selected vertex, subject to an impartiality constraint that the selection of a particular vertex is independent…

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity…

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

The Mostar index of a connected graph \(G\) is defined as \[ Mo(G)=\sum_{uv\in E(G)}\bigl|n_u(uv)-n_v(uv)\bigr|, \] where for an edge \(e=uv\), \(n_u(e)\) denotes the number of vertices of \(G\) that are closer to \(u\) than to \(v\). In…

The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph…

In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…

The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph…

This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.

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 $$f(n)\geq n+32t-1$$ for…

In a strong edge-coloring of a graph $G=(V,E)$, any two edges of distance at most $2$ get distinct colors. The strong chromatic index of $G$, denoted by $\chi_s'(G)$, is the minimum number of colors needed in a strong edge-coloring of $G$.…

A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…

The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been…

Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $\Delta > 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (such a…

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure…

We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$…

We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded…

The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp…

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e \in \Gamma(v)} f(e) \le 1$ for each $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The {\it fractional…