Related papers: A degree condition for diameter two orientability …
We prove that, for sufficiently large $n$, every graph of order $n$ with minimum degree at least $0.852n$ has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross to establish a bound of $0.9n$. By a…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let delta(H,n) denote the smallest integer t such that…
For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$,…
The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $\delta(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper,…
Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $\delta$ and $p$ - the order of a…
We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their…
Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…
Given a connected graph $G$ on $n$ vertices, with minimum degree $\delta\geq 2$ and girth at least $g \geq 4$, what is the maximum radius $r$ this graph can have? Erd\H{o}s, Pach, Pollack and Tuza established in the triangle-free case…
The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $\Delta$. This problem is well studied, and has been solved for plane graphs of low diameter in which every…
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph $G$, we examine the problem of assigning directions to each edge of $G$…
This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach…
A graph is said to be diameter-$k$-critical if its diameter is $k$ and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order $n$ has at most…
The "slope-number" of a graph $G$ is the minimum number of distinct edge slopes in a straight-line drawing of $G$ in the plane. We prove that for $\Delta\geq5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph with…
Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…
Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…
Galvin showed that for all fixed $\delta$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $\delta$ that admits the most independent sets is the complete bipartite graph $K_{\delta,n-\delta}$. He conjectured that except…
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the…
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity…
An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…