Forbidden Pairs of disconnected graphs for traceability and hamiltonicity

Abstract

In this paper, we firstly characterize all forbidden pairs $\{R,S\}$ for graphs with a spanning trail that are traceable and traceable graphs that are hamiltonian. There is no change of forbidden pairs for hamiltonicity if we impose a necessary condition of assumption that the graph is traceable; however, there is some difference of forbidden pairs for traceability if we impose a necessary condition that the graph has a spanning trail: different on two pairs of forbidden subgraphs $\{K_{1,3},Z_2\cup K_1\},\{K_{1,4},Z_1\}$ (where $Z_i$ is the graph obtained by identifying a vertex of a $K_3$ with an end-vertex of a $P_{i+1}$).

As a byproduct, we prove that if $G$ is a connected $\{K_{1,4},Z_1\}$-free graph, then every subgraph $G\left[T\right]$ induced by a trail $T$ is traceable and every subgraph $G\left[T\right]$ induced by a closed trail $T$ is either hamiltonian or $K_2\vee 3K_1$.