Related papers: Extremal Graphs Without 4-Cycles

A fundamental problem of extremal graph theory is to ask, 'What is the maximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They…

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…

In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle…

In this paper we show that for $r\geq 12$ and any sufficiently large $n$-vertex graph $G$ the number of $r$-edge-colorings of $G$ with no rainbow $K_4$ is at most $r^{ex(n,K_4)}$, where $ex(n,K_4)$ denotes the Tur\'{a}n number of $K_4$.…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chv\'{a}tal and Hanson (1976), and by Balachandran and Khare (2009). It follows from the structure of those extremal…

Let $G$ be a finite, connected graph. The eccentricity of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the arithmetic mean of the eccentricities of the vertices of $G$. We…

We show that any $3$-connected cubic plane graph on $n$ vertices, with all faces of size at most $6$, can be made bipartite by deleting no more than $\sqrt{(p+3t)n/5}$ edges, where $p$ and $t$ are the numbers of pentagonal and triangular…

We prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases it can be improved.…

We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs.

We present lower and upper bounds for the geometric-arithmetic index of unicyclic graphs and provide extremal graphs for the corresponding bounds.

We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for…

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the…

We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and…

A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex $v$ and every color $\alpha$, there are at most as many edges incident to $v$ colored with $\alpha$ as with all other colors.…

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…

This paper presents a characterization of edge-transitive graphs which are four regular and have girth four. This class consists of four infinite families plus four exceptional graphs.

A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known…

In 1975, Erd\H{o}s asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Tur\'an-type results can be used to prove an upper bound of…

The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of…