Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. conjectured [H. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571–576] that every 3-connected, essentially 4-connected line graph is Hamiltonian. In this note, we first show that the conjecture posed by Lai et al. is not tru...

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rc(G) is an NP-Hard problem, as well as that even deciding whether a graph has rc(G) =...

 The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $mathbb{A}(R...

Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C4), where C4 is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph i...

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123–129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs: an important one is that they are strongly connected. In [E. Cheng, M.J. Lipman, On the Day–Tripathi orientation of the star graphs: Connectivity, Inform. Process....

Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C), where C is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass-free line graph is ...

In [Discrete Math. 230 (2001), 133-141], it is shown that Tutte’s 3-flow conjecture that every 4-edge-connected graph has a nowhere zero 3-flow is equivalent to that every 4-edge-connected line graph has a nowhere zero 3-flow. We prove that every line graph of a 4-edgeconnected graph is Z3-connected. In particular, every line graph of a 4-edge-connected graph has a nowhere zero 3-flow.