Hamiltonian line graphs with local degree conditions

Abstract

Let N1,1,1 be the graph formed by attaching a pendant edge to each vertex of a triangle, and B1,2 be a graph obtained by attaching end vertices of two disjoint paths of lengths 1,2 to two vertices of a triangle. Broersma (1993) [2] and Čada et al. (2016) [3] conjectured that for a 2-connected claw-free simple graph G and for a fixed graph Γ∈{N1,1,1,B1,2,P6}, if δΓ(G)=min⁡{dG(v):dH(v)=1 for any induced subgraph H≅Γ in G}≥|V(G)|−23, then G is Hamiltonian. While Chen settles this conjecture recently, the following two results of the conjecture for 3-connected line graphs are proved.(i) For real numbers a,b with 0<a<1, there exists a family F(a,b) of finitely many nonsupereulerian graphs, such that for any 3-connected line graph H=L(G) of a simple graph G, if δN1,1,1(H)≥a|V(H)|+b, then either H is Hamiltonian or G is contractible to a member in F(a,b).(ii) Let H=L(G) be a 3-connected line graph of a simple graph G with |V(H)|≥116. If δN1,1,1(H)≥|V(H)|+510, then either H is Hamiltonian or G is isomorphic to the graph P(10)′, which is formed from the Petersen graph P(10) by attaching |V(H)|−1510 pendant edges to every vertex of P(10).