Related papers: Bounds for regular induced subgraphs of strongly r…

Albertson has defined the irregularity of a simple undirected graph $G=(V,E)$ as $ \irr(G) = \sum_{uv\in E}|d_G(u)-d_G(v)|,$ where $d_G(u)$ denotes the degree of a vertex $u \in V$. Recently, this graph invariant gained interest in the…

In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph…

Ramsey proved that for every positive integer $n$, every sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive…

A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \leq 4$, Pach and T\'oth proved a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph, which…

A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for…

Given a function $p : V(G)\to \mathbb N$ and an integer $k\ge 0$, define $p_k(G)$ as the number of vertices with $p(v)\ge k$. We say that $p_k(G)$ is bounded for all $\HH$-free graphs if there exists a constant $c=c(\HH)$ such that…

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

Twelve new strongly regular graphs with parameters (81,30,9,12) are found as graphs invariant under certain subgroups of the automorphism groups of the two previously known graphs that arise from 2-weight codes. One of these new graphs is…

We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for arbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, and Szemer\'edi resolved…

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random graph $G(n,p)$. Under the…

In 1972 Mader proved that every graph with average degree at least $4k$ has a $(k+1)$-connected subgraph with more than $2k$ vertices. We improve this bound by showing that the constant $4$ can be replaced by $3+\frac{1}{3}$; this bound is…

We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on…

For two given nonnegative integers $h$ and $k$, an $L(h,k)$-edge labeling of a graph $G$ is the assignment of labels $\{0,1, \cdots, n\}$ to the edges so that two edges having a common vertex are labeled with difference at least $h$ and two…

We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number of vertices in each subgraph is minimized. We prove NP-completeness of the problem, prove lower bounds, and give approximation…

We address the problem proposed by Chartrand, Erd\H{o}s and Oellermann (1988) about the existence of regular $K_3$-irregular graphs. We first establish bounds on the $K_3$-degrees of such graphs and use them to prove that there are no such…

Suppose a $k$-uniform hypergraph $H$ that satisfies a certain regularity instance (that is, there is a partition of $H$ given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities).…

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph…

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs,…