Hamiltonian cycles and 2-dominating induced cycles in claw-free graphs

Let G = (V, E) be a connected graph. For a vertex subset , G[S] is the subgraph of G induced by S. A cycle C (a path, respectively) is said to be an induced cycle (path, respectively) if G[V(C)] = C (G[V(P)] = P, respectively). The distance between a vertex x and a subgraph H of G is denoted by , where d(x, y) is the distance between x and y. A subgraph H of G is called 2-dominating if d(x, H) ≤ 2 for all . An induced path P of G is said to be maximal if there is no induced path P′ satisfying and . In this paper, we assume that G is a connected claw-free graph satisfying the following condition: for every maximal induced path P of length p ≥ 2 with end vertices u, v it holds:

Large cycles in 4-connected graphs

The first result states that every 4-connected graph G with minimum degree δ and connectivity κ either contains a cycle of length at least 4δ−κ−4 or every longest cycle in G is a dominating cycle. The second result states that any such graph under the additional condition δ≥α either contains a cycle of length at least 4δ−κ−4 or is hamiltonian, where α denotes the independence number of G. Both results are sharp in all respects.     

Dirac-type generalizations concerning large cycles in graphs

Bondy conjectured a common generalization of various results in hamiltonian graph theory concerning Hamilton and dominating cycles by introducing a notion of PD_λ-cycles (cycles that dominate all paths of lengths at least λ). We show that the minimum degree version of Bondy’s conjecture is true (along with the reverse version) if PD_λ-cycles are replaced by CD_λ-cycles (cycles that dominate all cycles of lengths at least λ). Fraisse proved a minimum degree generalization including a theorem of Nash-Williams for Hamilton cycles as a special case. We present the reverse version of this result including a theorem of Voss and Zuluaga as a special case. Two earlier less known results (due to the author) are crucial for the proofs of these results. All results are sharp in all respects. A number of possible similar generalizations are conjectured as well.     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

Hamiltonian cycles in k-partite graphs

Chen et al determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary $k$-partite graphs in that all parts have at most $n/2$ vertices (a necessary condition). To do this, we first prove a general result that both simplifies the process of checking whether a graph $G$ is a robust expander and gives useful structural information in the case when $G$ is not a robust expander. Then we use this result to prove that any $k$-partite graph satisfying the minimum degree condition is either a robust expander or else contains a Hamiltonian cycle directly.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

A fast parallel algorithm for finding Hamiltonian cycles in dense graphs

Suppose that 0<η<1 is given. We call a graph, G, on n vertices an η-Chvátal graph if its degree sequence d₁≤d₂≤?≤d_n satisfies: for k<n/2, d_k≤min{k+ηn,n/2} implies d_n−k−ηn≥n−k. (Thus for η=0 we get the well-known Chvátal graphs.) An -algorithm is presented which accepts as input an η-Chvátal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best -algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs (δ(G)≥n/2 where δ(G) is the minimum degree in G).     

Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G−S has a 1-factor for any SV(G) with |S|=n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with , then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G−S has a k-factor for any SV(G) with |S|=n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with , then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k2.     

Finding large cycles in Hamiltonian graphs

We show how to find in Hamiltonian graphs a cycle of length n^{Ω(1/loglogn)}=exp(Ω(logn/loglogn)). This is a consequence of a more general result in which we show that if G has a maximum degree d and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in O(n³) time a cycle in G of length k^Ω(1/logd). From this we infer that if G has a cycle of length k, then one can find in O(n³) time a cycle of length k^{Ω(1/(log(n/k)+loglogn))}, which implies the result for Hamiltonian graphs. Our results improve, for some values of k and d, a recent result of Gabow (2004) [11] showing that if G has a cycle of length k, then one can find in polynomial time a cycle in G of length . We finally show that if G has fixed Euler genus g and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in polynomial time a cycle in G of length f(g)k^Ω(1), running in time O(n²) for planar graphs.     

A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows.     

Hamiltonian cycles in 7-tough (P3 ∪ 2P1)-free graphs

The toughness of a noncomplete graph G is the maximum real number t such that the ratio of $|S|$ to the number of components of $G?S$ is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.     

On hamiltonian cycles in the prism over the odd graphs

The Kneser graph K(n, k) has as its vertex set all k‐subsets of an n‐set and two k‐subsets are adjacent if they are disjoint. The odd graph O_k is a special case of Kneser graph when n = 2k + 1. A long standing conjecture claims that O_k is hamiltonian for all k>2. We show that the prism over O_k is hamiltonian for all k even. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:177‐188, 2011     

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

In 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible.     

Traceability of line graphs

Brualdi and Shanny [R.A. Brualdi, R.F. Shanny, Hamiltonian line graphs, J. Graph Theory 5 (1981) 307-314], Clark [L. Clark, On hamitonian line graphs, J. Graph Theory 8 (1984) 303-307] and Veldman [H.J. Veldman, On dominating and spanning circuits in graphs, Discrete Math. 124 (1994) 229-239] gave minimum degree conditions of a line graph guaranteeing the line graph to be hamiltonian. In this paper, we investigate the similar conditions guaranteeing a line graph to be traceable. In particular, we show the following result: let G be a simple graph of order n and L(G) its line graph. If n is sufficiently large and, either ; or and G is almost bridgeless, then L(G) is traceable. As a byproduct, we also show that every 2-edge-connected triangle-free simple graph with order at most 9 has a spanning trail. These results are all best possible.     

关于Veldman等人的一个猜想

本文证明了下列源于Veldman等人的一个猜想;设G是阶为n≥13的1—tough图且对所有独立集x,y,z有d(x) d(y) d(z)≥(3n-14)/2,则G是哈米顿的。