Claw -free graphs and line graphs

The research of my dissertation is motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Tutte that every 4-edge-connected graph has a no-where-zero 3-flow. Towards the hamiltonian line graph problem, we proved that every 3-connected N2-locally connected claw-free graph is hamiltonian, which was conjectured by Ryjacek in 1990; that every 4-connected line graph of an almost claw free graph is hamiltonian connected, and that every triangularly connected claw-free graph G with |E( G)| ≥ 3 is vertex pancyclic. Towards the second conjecture, we proved that every line graph of a 4-edge-connected graph is Z 3-connected

On the s-Hamiltonian index of a graph

In modeling communication networks by graphs, the problem of designing s-fault-tolerant networks becomes the search for s-Hamiltonian graphs. This thesis is a study of the s-Hamiltonian index of a graph G.;A path P of G is called an arc in G if all the internal vertices of P are divalent vertices of G. We define l (G) = max{lcub}m : G has an arc of length m that is not both of length 2 and in a K3{rcub}. We show that if a connected graph G is not a path, a cycle or K1,3, then for a given s, we give the best known bound of the s-Hamiltonian index of the graph

A bound on connectivity of iterated line graphs

For simple connected graphs that are neither paths nor cycles, we define l(G) = max{m: G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path is a path whose internal vertices have degree two in G. Let G be a graph and Ln(G) be its n-th iterated line graph of G. We use (Formula Presented) and κ(G) for the essential edge connectivity and vertex connectivity of G, respectively. Let G be a simple connected graph that is not a path, a cycle or K1,3, with l(G) = l ≥ 1. We prove that (i) for integers (Formula Presented) (ii) for integers (Formula Presented). The bounds are best possible

Hamiltonicity in 3-connected claw-free graphs

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order ν and minimum degree δ ≥ (ν + 6)/10 is Hamiltonian for ν sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order ν, and if δ(H) ≥ (ν + 5)/10, then either H is Hamiltonian, or δ(H) = (ν + 5)/10 and the Ryjáček’s closure cl(H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding (ν − 15)/10 pendant edges at each vertex of P10

Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

AbstractLet G be a graph. For u,v∈V(G) with distG(u,v)=2, denote JG(u,v)={w∈NG(u)∩NG(v)|NG(w)⊆NG(u)∪NG(v)∪{u,v}}. A graph G is called quasi claw-free if JG(u,v)≠∅ for any u,v∈V(G) with distG(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected