Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)². We conjecture that linear growth of f suffices to imply hamiltonicity.     

Connectivity of the Mycielskian of a graph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This paper investigates the vertex-connectivity κ(μ(G)) and edge-connectivity κ^′(μ(G)) of μ(G) . We show that κ(μ(G))=min{δ(μ(G)),2κ(G)+1} and κ^′(μ(G))=δ(μ(G)).     

Bonds Intersecting Cycles In A Graph

Let G be a k-connected graph G having circumference c ≥ 2k. It is shown that for k ≥ 2, then there is a bond B which intersects every cycle of length c-k + 2 or greater.     

On the Toughness of a Graph

Abstract

In this paper, general results on the toughness of a graph are considered. Firstly the link between toughness and connectivity is explored and then results linking toughness and the parameters binding number and integrity are given. Further, the toughness of product graphs is discussed including general results for the lexicographic product. The paper concludes with some observations on toughness and hamiltoni-city.     

Contractible edges in non-separating cycles

An edge of ak-connected graph is said to bek-contractible if the contraction of the edge results in ak-connected graph. We prove that every triangle-freek-connected graphG has an induced cycleC such that all edges ofC arek-contractible and such thatG–V(C) is (k–3)-connected (k4). This result unifies two theorems by Thomassen [5] and Egawa et. al. [3].Dedicated to Professor Toshiro Tsuzuku on his sixtieth birthday     

Highly linked graphs

A graph with at least 2k vertices is said to bek-linked if, for any choices ₁,...,s _k ,t ₁,...,t _k of 2k distinct vertices there are vertex disjoint pathsP ₁,...,P _k withP _i joinings _i tot _i , 1ik. Recently Robertson and Seymour [16] showed that a graphG isk-linked provided its vertex connectivityk(G) exceeds . We show here thatk(G)22k will do.     

Disjoint shortest paths in graphs

It is an interesting problem that how much connectivity ensures the existence ofn disjoint paths joining givenn pairs of vertices, but to get a sharp bound seems to be very difficult. In this paper, we study how muchgeodetic connectivity ensures the existence ofn disjointgeodesics joining givenn pairs of vertices, where a graph is calledk-geodetically connected if the removal of anyk−1 vertices does not change the distance between any remaining vertices.     

On 3-skein isomorphisms of graphs

It is shown that a 3-skein isomorphism between 3-connected graphs with at least 5 vertices is induced by an isomorphism. These graphs have no loops but may be infinite and have multiple edges.     

Cycles through ten vertices in 3-connected cubic graphs

It is known that there exists a cycle through any nine vertices of a 3-connected cubic graphG. Here we show that if an edge is removed from such a graph, then there is still a cycle through any five vertices. Furthermore, we characterise the circumstances in which there fails to be a cycle through six. As corollaries we are able to prove that a 3-connected cubic graph has a cycle through any specified five vertices and one edge, and to classify the conditions under which it has a cycle through four chosen vertices and two edges. We are able to use the five and six vertex results to show that a 3-connected cubic graph has a cycle which passes through any ten given vertices if and only if the graph is not contractible to the Petersen graph in such a way that the ten vertices each map to a distinct vertex of the Petersen graph.     

Cycles through specified vertices of a graph

We prove that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG. Moreover, whenk≧3, each cycle ofG can be expressed as the sum of an odd number of cycles throughS. On the other hand, ifS is a set ofk vertices, these conclusions do not necessarily hold, and we characterize the exceptional cases. As corollaries, we establish the existence of odd and even cycles through specified vertices and deduce the existence of long odd and even cycles in graphs of high connectivity.     

A generalization of dirac’s theorem

LetG be an (r+2)-connected graph in which every vertex has degree at leastd and which has at least 2d-r vertices. Then, for any pathQ of lengthr and vertexy not onQ, there is a cycle of length at least 2d-r containing bothQ andy.     

Non-Separating Paths in 4-Connected Graphs

In 1975, Lovász conjectured that for any positive integer k, there exists a minimum positive integer f(k) such that, for any two vertices x, y in any f(k)-connected graph G, there is a path P from x to y in G such that G–V(P) is k-connected. A result of Tutte implies f(1) = 3. Recently, f(2) = 5 was shown by Chen et al. and, independently, by Kriesell. In this paper, we show that f(2) = 4 except for double wheels.Received October 17, 2003     

Path extendable graphs

A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K _k orK _k −e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13].     

Circuits Through Cocircuits In A Graph With Extensions To Matroids

We show that for any k-connected graph having cocircumference c*, there is a cycle which intersects every cocycle of size c*-k + 2 or greater. We use this to show that in a 2-connected graph, there is a family of at most c* cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley. A certain result known for cycles and cocycles in graphs is extended to matroids. It is shown that for a k-connected regular matroid having circumference c ≥ 2k if C₁ and C₂ are disjoint circuits satisfying r(C₁)+r(C₂)=r(C₁∪C₂), then |C₁|+|C₂|≤2(c-k + 1).     

Degree and local connectivity in digraphs

It is shown that there is a digraphD of minimum outdegree 12m and μ(x, y; D)=11m, but every digraphD of minimum outdegreen contains verticesx ≠y withλ(x, y; D)≧n−1, whereμ(x, y; D) andλ(x, y; D) denote the maximum number of openly disjoint and edge-disjoint paths, respectively.     

High Connectivity Keeping Sets In <Emphasis Type="Italic">n</Emphasis>-Connected Graphs

It is proved that for every positive integer k, every n-connected graph G of sufficiently large order contains a set W of k vertices such that G—W is (n-2)-connected. It is shown that this does not remain true if we add the condition that G(W) is connected.     

Coverings of the Vertices of a Graph by Small Cycles

Given a graph G with n vertices, we call c_k(G) the minimum number of elementary cycles of length at most k necessary to cover the vertices of G. We bound c_k(G) from the minimum degree and the order of the graph.     

A note on the girth of digraphs

Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than [n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding [n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.     

List Colouring When The Chromatic Number Is Close To the Order Of The Graph

Ohba has conjectured [7] that if G has 2 (G)+1 or fewer vertices then the list chromatic number and chromatic number of G are equal. In this short note we prove the weaker version of the conjecture obtained by replacing 2 (G)+1 by * This research was partially supported by DIMACS and by CNRS/NSF collaboration grant. Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.