We consider the derangements graph in which the vertices are permutations of f1 : : : ng. Two vertices are joined by an edge if the corresponding permutations diier in every position. The derangements graph is known to be hamil-tonian and it follows from a recent result of Jung that every pair of vertices is joined by a Hamilton path. We use this result to settle an open question, by showing th...

In this note we disprove the uniform shortest path routing conjec ture for vertex transitive graphs An in nite family of counterexam ples is given Let G be a connected graph on a vertex set V A routing in G is a set R fPuvj u v V V u vg of jV j jV j paths in G where each individual path Puv has initial vertex u and terminal vertex v We note that the paths Puv and Pvu may be di erent We say that...

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this article a lower and an upper bound for ψk of the rooted product graphs are presented. Two characterizations are given when those bounds are attained. Moreover ψ2 and ψ3 are exactly de...

A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χp(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χp(G) ≤ |V (G)| − α(G) + 1, where χ(G) and α(G) are the chromatic number and the independence number of G, re...

G. A. Dirac characterized chordal graphs as those in which minimal vertex separators always induce complete subgraphs. I generalize a traditional (2-)vertex separator to a k-vertex separator — meaning a set S of vertices whose removal puts k independent vertices into k separate components. Generalizing Dirac’s theorem, the {P5, 2P3}-free chordal graphs are the graphs in which minimal k-separato...

We show that the geodesic diameter of a polygonal domain with n vertices can be computed in O(n logn) time by considering O(n) candidate diameter endpoints; the endpoints are a subset of vertices of the overlay of shortest path maps from vertices of the domain.

Sum-free enumerative formulae are derived for several classes of rooted planar maps with no vertices of odd valency (eulerian maps) and with two vertices of odd valency (unicursal maps). As corollaries we obtain simple formulae for the numbers of unrooted eulerian and unicursal planar maps. Also, we obtain a sum-free formula for the number of rooted bi-eulerian (eulerian and bipartite) maps and...

We investigate the set of cycle lengths occurring in a hamiltonian graph with at least one or two vertices of large degree. We prove that in every case this set contains all the integers between 3 and some t, where t depends on the order of the graph and the degrees of vertices. c © 2002 Elsevier Science B.V. All rights reserved.

The eccentricity connectivity index of a molecular graph G is defined as (G) = aV(G) deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to other vertices of G and deg(a) is degree of vertex a. Here, we compute this topological index for some infinite classes of dendrimer graphs.

Let h be an arbitrary n-cycle in S n . Assume h ' is another n-cycle in S n . Then if σ = (h) 1 − h ' , σ is an element of A n , the alternating group of S n . From elementary group theory, every element of S n can be written as a product of (not necessarily disjoint) 3-cycles. Thus, h ' = hσ 1σ 2 ...σ r where each σ i (i = 1,2,...,r) is a 3-cycle. Defining h i = hσ 1σ 2 ...σ i , we now stipula...