Toughness and hamiltonicity in kk-trees

We consider toughness conditions that guarantee the existence of a hamiltonian cycle in $k$-trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to $\frac{7}{4}$. It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to $k$-trees for $k\ge 2$: Let $G$ be a $k$-tree. If $G$ has toughness at least $\frac{k+1}{3},$ then $G$ is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough $k$-trees for each $k\ge 3

The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this lower bound to $k/\lambda-1$ and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to $k/\lambda$. In this paper, we improve Brouwer's spectral bound when the toughness is small and we determine the exact value of the toughness for many strongly regular graphs attaining equality in the Hoffman ratio bound such as Lattice graphs, Triangular graphs, complements of Triangular graphs and complements of point-graphs of generalized quadrangles. For all these graphs with the exception of the Petersen graph, we confirm Brouwer's intuition by showing that the toughness equals $k/(-\lambda_{min})$, where $\lambda_{min}$ is the smallest eigenvalue of the adjacency matrix of the graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special issue dedicated to the "Applications of Graph Spectra in Computer Science" Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June 16-20, 201

On the Computational Complexity of Vertex Integrity and Component Order Connectivity

The Weighted Vertex Integrity (wVI) problem takes as input an $n$-vertex graph $G$, a weight function $w:V(G)\to\mathbb{N}$, and an integer $p$. The task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ plus the weight of a heaviest component of $G-X$ is at most $p$. Among other results, we prove that: (1) wVI is NP-complete on co-comparability graphs, even if each vertex has weight $1$; (2) wVI can be solved in $O(p^{p+1}n)$ time; (3) wVI admits a kernel with at most $p^3$ vertices. Result (1) refutes a conjecture by Ray and Deogun and answers an open question by Ray et al. It also complements a result by Kratsch et al., stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an $n$-vertex graph $G$, a weight function $w:V(G)\to \mathbb{N}$, and two integers $k$ and $l$, and the task is to decide if there exists a set $X\subseteq V(G)$ such that the weight of $X$ is at most $k$ and the weight of a heaviest component of $G-X$ is at most $l$. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We prove, among other results, that: (4) wCOC can be solved in $O(\min\{k,l\}\cdot n^3)$ time on interval graphs, while the unweighted version can be solved in $O(n^2)$ time on this graph class; (5) wCOC is W[1]-hard on split graphs when parameterized by $k$ or by $l$; (6) wCOC can be solved in $2^{O(k\log l)} n$ time; (7) wCOC admits a kernel with at most $kl(k+l)+k$ vertices. We also show that result (6) is essentially tight by proving that wCOC cannot be solved in $2^{o(k \log l)}n^{O(1)}$ time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the conference proceedings of ISAAC 201

Conditions for minimally tough graphs

Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded from the class of minimally tough graphs. In this paper, we consider the opposite question, namely which induced subgraphs, if any, must necessarily be present in each minimally $t$-tough graph. Katona and Varga showed that for any rational number $t \in (1/2,1]$, every minimally $t$-tough graph contains a hole. We complement this result by showing that for any rational number $t>1$, every minimally $t$-tough graph must contain either a hole or an induced subgraph isomorphic to the $k$-sun for some integer $k \ge 3$. We also show that for any rational number $t > 1/2$, every minimally $t$-tough graph must contain either an induced $4$-cycle, an induced $5$-cycle, or two independent edges as an induced subgraph

Polyhedra with few 3-cuts are hamiltonian

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each $k \ge 6$ non-hamiltonian polyhedra with $k$ 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl