Hamiltonian results in K1,3-free graphs

There have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K_1,3-free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K_1,3-free graphs.     

On connected factors inK 1,3-free graphs

A graph is said to beK _1,3-free if it contains noK _1,3 as an induced subgraph. It is shown in this paper that every 2-connectedK _1,3-free graph contains a connected [2,3]-factor. We also obtain that every connectedK _1,3-free graph has a spanning tree with maximum degree at most 3. This research is partially supported by the National Natural Sciences Foundation of China and by the Natural Sciences Foundation of Shandong Province of China.     

Regular factors in K1,3-free graphs

We show that every connected K_1,3-free graph with minimum degree at least 2k contains a k-factor and construct connected K_1,3-free graphs with minimum degree k + 0(√k) that have no k-factor.     

Longest paths and cycles in K1,3-free graphs

In this article we show that the standard results concerning longest paths and cycles in graphs can be improved for K_1,3-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K_1,3-free graphs.     

Every 3-connected {K 1,3,N 3,3,3}-free graph is Hamiltonian

For non-negative integers i, j and k, let N _i,j,k be the graph obtained by identifying end vertices of three disjoint paths of lengths i, j and k to the vertices of a triangle. In this paper, we prove that every 3-connected {K _1,3,N _3,3,3}-free graph is Hamiltonian. This result is sharp in the sense that for any integer i > 3, there exist infinitely many 3-connected {K _1,3,N _i,3,3}-free non-Hamiltonian graphs.     

Connected,locally 2-connected,K1,3-free graphs are panconnected

A graph G is locally n-connected, n ≥ 1, if the subgraph induced by the neighborhood of each vertex is n-connected. We prove that every connected, locally 2-connected graph containing no induced subgraph isomorphic to K_1,3 is panconnected.     

Hamiltonian circuits in Cayley graphs

The following result is proved: If either G is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected Cayley graph of G is hamiltonian.     

Hamiltonian N2‐locally connected claw‐free graphs

A graph G is N₂‐locally connected if for every vertex ν in G, the edges not incident with ν but having at least one end adjacent to ν in G induce a connected graph. In 1990, Ryjá?ek conjectured that every 3‐connected N₂‐locally connected claw‐free graph is Hamiltonian. This conjecture is proved in this note. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 142–146, 2005     

Finding Hamiltonian circuits in quasi-adjoint graphs

This paper is motivated by a method used for DNA sequencing by hybridization presented in [Jacek Blazewicz, Marta Kasprzak, Computational complexity of isothermic DNA sequencing by hybridization, Discrete Appl. Math. 154 (5) (2006) 718–729]. This paper presents a class of digraphs: the quasi-adjoint graphs. This class includes the ones used in the paper cited above. A polynomial recognition algorithm in O(n³), as well as a polynomial algorithm in O(n²+m²) for finding a Hamiltonian circuit in these graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a path with three vertices) are discussed.     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Dominating sets and hamiltonicity inK 1,3-free graphs

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-011-1486).     

A generalization of Dirac’s theorem for K(1,3)-free graphs

It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3)-free graph of ordern with the cardinality of the union of the neighborhoods of each pair of vertices at least (n+1)/3, thenG is hamiltonian. We also investigate several other related properties inK(1,3)-free graphs such as traceability, hamiltonian-connectedness, and pancyclicity. Partially Supported by O. N. R. Contract Number N00014-88-K-0070. Partially Supported by O. N. R. Contract Number N00014-85-K-0694.     

The square of a connected S(K1,3)-free graph is vertex pancyclic

We prove the conjecture of Gould and Jacobson that a connected S(K_1,3)-free graph has a vertex pancyclic square. Since S(K_1,3)² is not vertex pan-cyclic, this result is best possible.     

A necessary and sufficient condition for connected,locally k-connected k1,3-free graphs to be k-hamiltonian

We prove the following conjecture of Broersma and Veldman: A connected, locally k-connected K_1,3-free graph is k-hamiltonian if and only if it is (k + 2)-connected (K ? 1).     

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f _s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K _s }}, where the minimum is taken over all K _t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n ^1/2+o(1)) ≤ f _s,s+1(n) ≤ O(n ^1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f _s,s+1(n) ≤ O(n ^2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n ^1/2−ɛ ) ≤ f _s,s+k (n) ≤ O(n ^1/2+ɛ . In addition, we also discuss some connections between the function f _s,t and vertex Folkman numbers.