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Title:
Toughness, forbidden subgraphs, and Hamiltonian-connected graphs
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Discussiones Mathematicae Graph Theory 42(1) (2022) 187-196
Received: 2019-05-27 , Revised: 2019-08-30 , Accepted: 2019-08-30 , Available online: 2019-11-02 , https://doi.org/10.7151/dmgt.2247
Abstract:
A graph $G$ is called Hamilton-connected if for every pair of distinct vertices
$\{u,v\}$ of $G$ there exists a Hamilton path in $G$ that connects $u$ and $v$.
A graph $G$ is said to be $t$-tough if $t\cdot \omega(G-X) \leq |X|$ for all
$X\subseteq V(G)$ with $\omega(G-X)>1$. The toughness of $G$, denoted $\tau(G)$,
is the maximum value of $t$ such that $G$ is $t$-tough (taking
$\tau(K_n)=\infty$ for all $n\geq1$). It is known that a Hamilton-connected
graph $G$ has toughness $\tau(G)>1$, but that the reverse statement does not
hold in general. In this paper, we investigate all possible forbidden subgraphs
$H$ such that every $H$-free graph $G$ with $\tau(G)>1$ is Hamilton-connected.
We find that the results are completely analogous to the {Hamiltonian} case:
every graph $H$ such that any 1-tough $H$-free graph is {Hamiltonian} also ensures
that every $H$-free graph with toughness larger than one is Hamilton-connected.
And similarly, there is no other forbidden subgraph having this property, except
possibly for the graph $K_1\cup P_4$ itself. We leave this as an open case.
Keywords:
toughness, forbidden subgraph, Hamiltonian-connected graph, Hamiltonicity
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