Related papers: Enumeration and limit laws of series-parallel grap…

We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$…

An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges…

We show that any n-vertex graph without even cycles of length at most 2k has at most 1/2(n^{1 + 1/k}) + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k = 2,3,5.

We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…

We prove a $pre$-$asymptotic$ bound on the total variation distance between the uniform distribution over two types of undirected graphs with $n$ nodes. One distribution places a prescribed number of $k_T$ triangles and $k_S$ edges not…

For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is then…

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the…

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good…

One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish…

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=\omega(N^{1/3}\ln^{2} N)$. This almost completely fills the gap…

We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete…

For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set…

One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when…

In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…

For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian…

We prove an asymptotic formula for the number of Eulerian circuits for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error $O(n^{-1/2+\varepsilon})$, where…

Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$…

As proved by Kahn, the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph $G$ we have $\chi(G) \leq (1+o(1))\chi_f(G)$. We extend this result to quasi-line graphs, an important…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the…