Related papers: On vertex types of graphs

In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging…

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest…

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, nut graphs have seven or more vertices;…

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in…

In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…

A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected…

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex.…

Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a variant of this classical problem in which the position of each…

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that…

We get central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on distributions of the weights.

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…

We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.

For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and…

Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs…

Suppose that $G$ is a simple, vertex-labeled graph and that $S$ is a multiset. Then if there exists a one-to-one mapping between the elements of $S$ and the vertices of $G$, such that edges in $G$ exist if and only if the absolute…

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…