相关论文: Graphs with restricted valency and matching number

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.

A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…

For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $\chi'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges…

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…

Settling Kahn's conjecture (2001), we prove the following upper bound on the number $i(G)$ of independent sets in a graph $G$ without isolated vertices: \[ i(G) \le \prod_{uv \in E(G)} i(K_{d_u,d_v})^{1/(d_u d_v)}, \] where $d_u$ is the…

A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary…

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly…

We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an…

For a graph $G$, we show that if $mad(G)<m$, then $\chi'_\ell(G)\leq \Delta+1$ where $m$ depends upon $\Delta$ and $\chi'_\ell(G)$ is the list-chromatic index of $G$. When $\Delta\leq 20$ the value of $m$ is close to $\frac{1}{2}\Delta$,…

Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a…

We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be…

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…

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…

Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs…

In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and $\delta(G)$…

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…