Cycles in 2-Factors of Balanced Bipartite Graphs

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with . Revised: May 7, 1999     

6-regular Cayley graphs on abelian groups of odd order are hamiltonian decomposable

Alspach conjectured that any 2k-regular connected Cayley graph on a finite abelian group A has a hamiltonian decomposition. In this paper, the conjecture is shown true if k=3, and the order of A is odd.     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

On two-factors of bipartite regular graphs

The set of two-factors of a bipartite k-regular graph, k>2, spans the cycle space of the graph. In addition, a new non-hamiltonian 3-connected bicubic graph on 92 vertices is constructed.     

Hamilton decompositions of graphs with primitive complements

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1≤k<Δ(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions for the existence of a 2x-regular graph G on n vertices which:

1.

has a Hamilton decomposition, and

2.

has a complement in K_n that is primitive.

This extends the conditions studied by Hoffman, Rodger, and Rosa [D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamiltonian cycles, J. Combin. Theory Ser. B 57 (1) (1993) 69-76] who considered maximal sets of Hamilton cycles and 2-factors. It also sheds light on construction approaches to the Hamilton-Waterloo problem.     

Hamilton decompositions of some line graphs

The main result of this paper completely settles Bermond's conjecture for bipartite graphs of odd degree by proving that if G is a bipartite (2k + 1)-regular graph that is Hamilton decomposable, then the line graph, L(G), of G is also Hamilton decomposable. A similar result is obtained for 5-regular graphs, thus providing further evidence to support Bermond's conjecture. © 1995 John Wiley & Sons, Inc.     

The virtual k-factorability of graphs

It is shown that if a connected graph G admits a finite covering that has a k-factor, then there is a 1- or 2-fold covering of G that has a k-factor and that such a covering can be given as a canonical bipartite graph associated with G.     

Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.     

Degree Sum Conditions for Hamiltonicity on k-Partite Graphs

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graph G has order p and minimum degree at least \(\frac{p} {2}\) then G is hamiltonian. Moon and Moser showed that if G is a balanced bipartite graph (the two partite sets have the same order) with minimum degree more than \(\frac{p} {4}\) then G is hamiltonian. Recently their idea is generalized to k-partite graphs by Chen, Faudree, Gould, Jacobson, and Lesniak in terms of minimum degrees. In this paper, we generalize this result in terms of degree sum and the following result is obtained: Let G be a balanced k-partite graph with order kn. If for every pair of nonadjacent vertices u and v which are in different parts $$d(u) + d(v) > \left\{ {\begin{array}{*{20}c} {\left( {k - \frac{2} {{k + 1}}} \right)n} & {if k is odd} \\ {\left( {k - \frac{4} {{k + 2}}} \right)n} & {if k is even} \\ \end{array} } \right.,$$ then G is hamiltonian.     

On low degree k-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v₁,…,v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k-orderedness of a graph implies (k-1)-connectivity, they posed the question of the existence of low degree k-ordered hamiltonian graphs. We construct an infinite family of graphs, which we call bracelet graphs, that are (k-1)-regular and are k-ordered hamiltonian for odd k. This result provides the best possible answer to the question of the existence of low degree k-ordered hamiltonian graphs for odd k. We further show that for even k, there exist no k-ordered bracelet graphs with minimum degree k-1 and maximum degree less than k+2, and we exhibit an infinite family of bracelet graphs with minimum degree k-1 and maximum degree k+2 that are k-ordered for even k. A concept related to k-orderedness, namely that of k-edge-orderedness, is likewise strongly related to connectivity properties. We study this relation and give bounds on the connectivity necessary to imply k-(edge-)orderedness properties.     

Hamiltonian decomposition of Cayley graphs of ordersp 2 andpq

In this paper, it is proved that any connected Cayley graph on an abelian group of order pq orp ² has a hamiltonian decomposition, wherep andq are odd primes. This result answers partially a conjecture of Alspach concerning hamiltonian decomposition of 2k-regular connected Cayley graphs on abelian groups.     

Non-separating 2-factors of an even-regular graph

For a 2-factor F of a connected graph G, we consider G−F, which is the graph obtained from G by removing all the edges of F. If G−F is connected, F is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r-regular connected graph G to have such a 2-factor. As a result, we show that a 2r-regular connected graph G has a non-separating 2-factor whenever the number of vertices of G does not exceed 2r²+r.     

On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.     

Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk – 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank – 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr 0, stillG – S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.     

Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each 2-factor of the graph is always even or always odd. We proved (Abreu et?al., J Comb Theory B 98:432–442, 2008) that the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graph of girth 4 is K _3,3, and conjectured (Abreu et?al., 2008, Conjecture 3.6) that the only essentially 4-edge-connected cubic bipartite graphs are K _3,3, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations n ₃ due to Martinetti (1886) in which all symmetric configurations n ₃ can be obtained from an infinite set of so called irreducible configurations (Martinetti, Annali di Matematica Pura ed Applicata II 15:1–26, 1888). The list of irreducible configurations has been completed by Boben (Discret Math 307:331–344, 2007) in terms of their irreducible Levi graphs. In this paper we characterize irreducible pseudo 2-factor isomorphic cubic bipartite graphs proving that the only pseudo 2-factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture.     

Maximum matchings in a regular graph of specified connectivity and bounded order

Upper bounds are placed on the order of a k-regular m-connected graph G that produce a lower bound on the number of independent edges in G. As a corollary, we obtain the order of a smallest k-regular m-connected graph which has no 1-factor.     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

A sufficient condition for a bipartite graph to have a k-factor

P. Katerinis obtained a sufficient condition for the existence of a 2-factor in a bipartite graph, in the spirit of Hall's theorem. We show a sufficient condition for the existence of a k-factor in a bipartite graph, as a generalization of Katerinis's theorem and Hall's theorem.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.