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.     

On the connectivity of randomm-orientable graphs and digraphs

We consider graphs and digraphs obtained by randomly generating a prescribed number of arcs incident at each vertex. We analyse their almost certain connectivity and apply these results to the expected value of random minimum length spanning trees and arborescences. We also examine the relationship between our results and certain results of Erdős and Rényi.     

The least eigenvalue of graphs with given connectivity

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of G are those of A(G). In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity.     

Partition of graphs with condition on the connectivity and minimum degree

C. Thomassen and M. Szegedy proved the existence of a functionf(s, t) such that the points of anyf(s, t)-connected graph have a decomposition into two non-empty sets such that the subgraphs induced by them ares-connected andt-connected, respectively. We prove, thatf(s, t) ≦ 4s+4t − 13 and examine a similar problem for the minimum degree.     

Additivity of the crossing number of graphs with connectivity 2

We prove that the crossing number of graphs with connectivity 2 has in certain cases an additive property analogous to that of crossing number of graphs with connectivity ≤1.     

Pairs of edge disjoint Hamiltonian circuits in 5-connected planar graphs

Summary A variety of examples of 4-connected 4-regular graphs with no pair of disjoint Hamiltonian circuits were constructed in response to Nash-Williams conjecture that every 4-connected 4-regular graph is Hamiltonian and also admits a pair of edge-disjoint Hamiltonian circuits. Nash-Williams's problem is especially interesting for planar graphs since 4-connected planar graphs are Hamiltonian. Examples of 4-connected 4-regular planar graphs in which every pair of Hamiltonian circuits have edges in common are included in the above mentioned examples.B. Grünbaum asked whether 5-connected planar graphs always admit a pair of disjoint Hamiltonian circuits. In this paper we introduce a technique that enables us to construct infinitely many examples of 5-connected planar graphs, 5-regular and non regular, in which every pair of Hamiltonian circuits have edges in common.     

Proof of a conjecture of T. Gallai concerning connectivity properties of colour-critical graphs

Tibor Gallai made the following conjecture. LetG be ak-chromatic colour-critical graph. LetL denote the set of those vertices ofG having valencyk−1 and letH be the rest ofV(G). Then the number of components induced byL is not less than the number of components induced byH, providedL ≠ 0. We prove this conjecture in a slightly generalized form. Dedicated to Tibor Gallai on his seventieth birthday     

Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity

In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity.     

The algebraic connectivity of lollipop graphs

Let C_n,g be the lollipop graph obtained by appending a g-cycle C_g to a pendant vertex of a path on n-g vertices. In 2002, Fallat, Kirkland and Pati proved that for and g?4, α(C_n,g)>α(C_n,g-1). In this paper, we prove that for g?4, α(C_n,g)>α(C_n,g-1) for all n, where α(C_n,g) is the algebraic connectivity of C_n,g.     

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.     

Contractible edges in triangle-free graphs

An edge of a graph is calledk-contractible if the contraction of the edge results in ak-connected graph. Thomassen [5] proved that everyk-connected graph of girth at least four has ak-contractible edge. In this paper, we study the distribution ofk-contractible edges in triangle-free graphs and show the following: Whenk≧2, everyk-connected graph of girth at least four and ordern≧3k, hasn+(3/2)k ²-3k or morek-contractible edges.     

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.     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d⁺(u),d^-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when

λ(u,v)=min{d⁺(u),d^-(v)}     

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.     

The sextet construction for cubic graphs

We show how to construct cubic graphs which have automorphism groups acting regularly on thes-arcs,s=4 or 5.     

Dense graphs and edge reconstruction

By a well-known result of Nash-Williams if a graphG is not edge reconstructible, then for all ,|A||E(G)| mod 2 we have a permutation ofV(G) such thatE(G)E(G)=A. Here we construct infinitely many graphsG having this curious property and more than edges.Research (partially) supported by Hungarian National Foundation for Scientific Research Grant No.T016389.     

Decomposition of 3-connected graphs

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K _3,n for somen3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

Bounds on total domination in claw-free cubic graphs

A set S of vertices in a graph G is a total dominating set, denoted by TDS, of G if every vertex of G is adjacent to some vertex in S (other than itself). The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). If G does not contain K_1,3 as an induced subgraph, then G is said to be claw-free. It is shown in [D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek, P.C.B. Lam, S. Seager, B. Wei, R. Yuster, Some remarks on domination, J. Graph Theory 46 (2004) 207-210.] that if G is a graph of order n with minimum degree at least three, then γ_t(G)?n/2. Two infinite families of connected cubic graphs with total domination number one-half their orders are constructed in [O. Favaron, M.A. Henning, C.M. Mynhardt, J. Puech, Total domination in graphs with minimum degree three, J. Graph Theory 34(1) (2000) 9-19.] which shows that this bound of n/2 is sharp. However, every graph in these two families, except for K₄ and a cubic graph of order eight, contains a claw. It is therefore a natural question to ask whether this upper bound of n/2 can be improved if we restrict G to be a connected cubic claw-free graph of order at least 10. In this paper, we answer this question in the affirmative. We prove that if G is a connected claw-free cubic graph of order n?10, then γ_t(G)?5n/11.