Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words

2-factors in 32\frac{3}{2}-tough plane triangulations

In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is hamiltonian. This result implies that every more than $\frac{3}{2}$-tough planar graph on at least three vertices is hamiltonian and so has a 2-factor. Owens in 1999 constructed non-hamiltonian maximal planar graphs of toughness arbitrarily close to $\frac{3}{2}$. In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly $\frac{3}{2}$ is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by showing that every maximal $\frac{3}{2}$-tough planar graph on at least three vertices has a 2-factor

Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

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

On Hamiltonicity of {claw, net}-free graphs

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an edge e to have (1)a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian s- and st-paths containing edge e when G has connectivity one, and (4) a Hamiltonian cycle containing e when G is 2-connected. These results imply that a connected {claw, net}-free graph has a Hamiltonian path and a 2-connected {claw, net}-free graph has a Hamiltonian cycle [D. Duffus, R.J. Gould, M.S. Jacobson, Forbidden Subgraphs and the Hamiltonian Theme, in The Theory and Application of Graphs (Kalamazoo, Mich., 1980$), Wiley, New York (1981) 297--316.] Our proofs of (1)-(4) are shorter than the proofs of their corollaries in [D. Duffus, R.J. Gould, M.S. Jacobson] and provide polynomial-time algorithms for solving the corresponding Hamiltonicity problems. Keywords: graph, claw, net, {claw, net}-free graph, Hamiltonian path, Hamiltonian cycle, polynomial-time algorithm.Comment: 9 page

Supereulerian Properties in Graphs and Hamiltonian Properties in Line Graphs

Following the trend initiated by Chvatal and Erdos, using the relation of independence number and connectivity as sufficient conditions for hamiltonicity of graphs, we characterize supereulerian graphs with small matching number, which implies a characterization of hamiltonian claw-free graph with small independence number.;We also investigate strongly spanning trailable graphs and their applications to hamiltonian connected line graphs characterizations for small strongly spanning trailable graphs and strongly spanning trailable graphs with short longest cycles are obtained. In particular, we have found a graph family F of reduced nonsupereulerian graphs such that for any graph G with kappa\u27(G) ≥ 2 and alpha\u27( G) ≤ 3, G is supereulerian if and only if the reduction of G is not in F..;We proved that any connected graph G with at most 12 vertices, at most one vertex of degree 2 and without vertices of degree 1 is either supereulerian or its reduction is one of six exceptional cases. This is applied to show that if a 3-edge-connected graph has the property that every pair of edges is joined by a longest path of length at most 8, then G is strongly spanning trailable if and only if G is not the wagner graph.;Using charge and discharge method, we prove that every 3-connected, essentially 10-connected line graph is hamiltonian connected. We also provide a unified treatment with short proofs for several former results by Fujisawa and Ota in [20], by Kaiser et al in [24], and by Pfender in [40]. New sufficient conditions for hamiltonian claw-free graphs are also obtained