Neighborhood intersections and Hamiltonicity in almost claw-free graphs

AbstractLet G be a graph. The partially square graph G∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv∉E(G), and there is some w∈N(u)∩N(v), such that N(w)⊆N(u)∪N(v)∪{u,v}. Let t>1 be an integer and Y⊆V(G), denote n(Y)=|{v∈V(G)|miny∈Y{distG(v,y)}⩽2}|,It(G)={Z|Z is an independent set of G,|Z|=t}. In this paper, we show that a k-connected almost claw-free graph with k⩾2 is hamiltonian if ∑z∈Zd(z)⩾n(Z)−k in G for each Z∈Ik+1(G∗), thereby solving a conjecture proposed by Broersma, Ryjác̆ek and Schiermeyer. Zhang's result is also generalized by the new result

Eulerian subgraphs and Hamiltonicity of claw -free graphs

Let C(l, k) denote the class of 2-edge-connected graphs of order n such that a graph G ∈ C(l, k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G - S has order at least n-kl . We prove that If G ∈ C(6, 0), then G is supereulerian if and only if G cannot be contracted to K2,3, K 2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. Previous results by Catlin and Li, and by Broersma and Xiong are extended.;We also investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We determine several extremal bounds for planar graphs to be supereulerian.;Kuipers and Veldman conjectured that any 3-connected claw-free graph with order n and minimum degree delta ≥ n+610 is Hamiltonian for n sufficiently large. We prove that if H is a 3-connected claw-free graph with sufficiently large order n, and if delta(H) ≥ n+510 , then either H is hamiltonian, or delta( H) = n+510 and the Ryjac˘ek\u27s closure cl( H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding n-1510 pendant edges at each vertex of P10

Z3-connectivity of 4-edge-connected 2-triangular graphs

AbstractA graph G is k-triangular if each edge of G is in at least k triangles. It is conjectured that every 4-edge-connected 1-triangular graph admits a nowhere-zero Z3-flow. However, it has been proved that not all such graphs are Z3-connected. In this paper, we show that every 4-edge-connected 2-triangular graph is Z3-connected. The result is best possible. This result provides evidence to support the Z3-connectivity conjecture by Jaeger et al that every 5-edge-connected graph is Z3-connected

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

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