Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the number of edges of $C\!D(k,q)$, then $e =Ω(v^{1+ {1\over {k-t+1}}})$. These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order $v$ and girth at least $g$, $ g\ge 5$, $g \not= 11,12$. For $g\ge 24$, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for $5\le g\le 23$, $g\not= 11,12$, it improves on or ties existing bounds.
In this note we answer a question of Hurlbert about pebbling in graphs of high girth. Specifically we show that for every g there is a Class 0 graph of girth at least g. The proof uses the so-called Erdos construction and employs a recent result proved by Czygrinow, Hurlbert, Kierstead and Trotter. We also use the Czygrinow et al. result to prove that Graham's pebbling product conjecture holds for dense graphs. Finally, we consider a generalization of Graham's conjecture to thresholds of graph sequences and find reasonably tight bounds on the pebbling threshold of the sequence of d-dimensional…
We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.
A variant of the Bourgain-Gamburd machine without using any girth bounds is obtained. Also, we find series of applications of %our result the Bourgain-Gamburd machine to problems of Additive Combinatorics, Number Theory and Probability.
We prove that every bipartite graph of sufficiently large average degree has either a $K_{t,t}$-subgraph or an induced subgraph of average degree at least $t$ and girth at least $6$. We conjecture that "$6$" can be replaced by "$k$", which strengthens a conjecture of Thomassen. In support of this conjecture, we show that it holds for regular graphs.
Impact of local girth on the S-packing coloring of k-saturated subcubic graphs
University of Sciences and Arts in Lebanon（黎巴嫩科学与艺术学院）；KALMA Laboratory, Faculty of Sciences, Lebanese University（黎巴嫩大学理学院 KALMA 实验室）；Al Maaref University（阿尔马雷夫大学）；Lebanese International University (LIU)（黎巴嫩国际大学）
The strong fractional choice number of a graph $G$ is the infimum of those real numbers $r$ such that $G$ is $(\lceil rm \rceil, m)$-choosable for every positive integer $m$. The strong fractional choice number of a family ${\cal G}$ of graphs is the supremum of the strong fractional choice number of graphs in ${\cal G}$. We denote by ${\cal{Q}}_k$ the class of series-parallel graphs with girth at least $k$. This paper proves that for $k=4q-1, 4q,4q+1, 4q+2$, the strong fractional number of ${\cal{Q}}_k$ is exactly $2+ \frac{1}{q}$.
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by László Lovász and his coauthors. We prove that spectra of graphs, and also some important graph parameters such as numbers of colorings or matchings, behave well in convergent graph sequences. Special attention is given to graph sequences of large essential girth, for which asymptotics of coloring numbers are explicitly calculated. We also treat numbers of matchings in approximately regular graphs. We introduce tentative lim…
Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H^2. Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph. The graph introduced in their reduction is a graph that contains many triangles and is relatively dense. Farzad et al. proved the NP-completeness for finding a square root for girth 4 while they gave a polynomial time algorithm for comput…
We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Turán's theorem. We also demonstrate (nonconstructively) the existence of dense $k$-critical graphs avoiding all odd cycles of length $\leq \ell$ for any $\ell$ and any $k\geq 4$, again with a best possible density of $>(\frac{1}{4} -\varepsilon)n^2$ edges for $k\geq 6$. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the c…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. A graph is called 2-$degenerate$ if any of its induced subgraph has a vertex of degree at most 2. The class of 2-$degenerate graphs$ properly conta in $series$-$parallel graphs$, $outerplanar graphs$, \emph{non-regular subcubic graphs}, \emph{planar graphs of girth at least 6} and \emph{circle graphs of girth at least 5} as subclasse…
For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function $ex(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of $ex(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.
We study a variant of intersection representations with unit balls, that is, unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far sets, the goal is to represent the vertices of the graph by unit balls so that the balls representing two adjacent vertices intersect if and only if the corresponding edge is near. We consider the problem in the plane and prove that it is NP-hard to decide whether such a representation exists for a given edge-partition. On the other hand, every series-parallel graph admits such a r…
Asymptotics for graphically divergent series: dense digraphs and 2-SAT formulae
LaBRI, Université de Bordeaux（波尔多大学）；LIPN, Université Sorbonne Paris Nord（索邦巴黎北大学）；IMB, Université de Bourgogne（勃艮第大学）；LIB, Université de Bourgogne（勃艮第大学）；LIP6, Sorbonne Université（索邦大学）
We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge 1$. We prove that $G$ has a spanning tree with at least $α_{g,k}(v(G)-k-2)+2$ leaves, where $α_{g,k}= {[{g+1\over2}]\over [{g+1\over2}](k+3)+1}$ for $k<g-2$; $α_{g,k}= {g-2\over (g-1)(k+2)}$ for $k\ge g-2$. We present infinite series of examples showing that all these bounds are exact.
Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is th…
In this note, we show how to obtain a "characteristic power series" of graphons -- infinite limits of dense graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a new characterization of graph quasi-randomness and another perspective on spectral theory for graphons, a complete description of the function in terms of the spectrum of the graphon as a self-adjoint kernel operator. Interestingly, while we apply a standard regularization to classical determinants, it is unclear how necessary this is.
