On Hamiltonian cycles in balanced kk-partite graphs

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least $$\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases} \lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k} & : k \text{ odd }\\ \frac{n}{2}+\lfloor\frac{n+2}{k+2}\rfloor-\frac{n}{k} & : k \text{ even } \end{cases},$$ then $G$ has a Hamiltonian cycle unless $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$. In the case where $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$, we can characterize the graphs which do not have a Hamiltonian cycle and see that $\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}+1$ suffices. This result is tight for all $k\geq 2$ and $n\geq 3$ divisible by $k$.Comment: 14 pages, 2 figures. Minor update

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given

Partitioning 3-colored complete graphs into three monochromatic cycles

We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible

Chorded pancyclicity in kk-partite graphs

We prove that for any integers $p\geq k\geq 3$ and any $k$-tuple of positive integers $(n_1,\ldots ,n_k)$ such that $p=\sum _{i=1}^k{n_i}$ and $n_1\geq n_2\geq \ldots \geq n_k$, the condition $n_1\leq {p\over 2}$ is necessary and sufficient for every subgraph of the complete $k$-partite graph $K(n_1,\ldots ,n_k)$ with at least $${{4 -2p+2n_1+\sum _{i=1}^{k} n_i(p-n_i)}\over 2}$$ edges to be chorded pancyclic. Removing all but one edge incident with any vertex of minimum degree in $K(n_1,\ldots ,n_k)$ shows that this result is best possible. Our result implies that for any integers, $k\geq 3$ and $n\geq 1$, a balanced $k$-partite graph of order $kn$ with has at least ${{(k^2-k)n^2-2n(k-1)+4}\over 2}$ edges is chorded pancyclic. In the case $k=3$, this result strengthens a previous one by Adamus, who in 2009 showed that a balanced tripartite graph of order $3n$, $n \geq 2$, with at least $3n^2 - 2n + 2$ edges is pancyclic.Comment: The manuscript was submitted with title {\it A note on pancyclicity of $k$-partite graphs} and accepted with the title {\it Chorded pancyclicity in $k$-partite graphs.} This version corresponds to the paper accepted for publication in Graphs. Combi

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented. The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable. The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case. Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory

New Sufficient Conditions for Hamiltonian Paths

A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph

On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs

We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.Comment: Comments are welcome

Multiply Balanced Edge Colorings of Multigraphs

In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out among the vertices in ways that are fair with respect to several notions of balance (such as between pairs of vertices, degrees of vertices in the both graph and in each color class, etc). The connectivity of color classes is also addressed. Most results in the literature on amalgamations focus on the disentangling of amalgamated complete graphs and complete multipartite graphs. Many such results follow as immediate corollaries to the main result in this paper, which addresses amalgamations of graphs in general, allowing for example the final graph to have multiple edges. A new corollary of the main theorem is the settling of the existence of Hamilton decompositions of the family of graphs $K(a_1,\dots, a_p;\lambda_1, \lambda_2)$; such graphs arose naturally in statistical settings.Comment: 21 pages, 2 figure

Problem collection from the IML programme: Graphs, Hypergraphs, and Computing

This collection of problems and conjectures is based on a subset of the open problems from the seminar series and the problem sessions of the Institut Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem contributor has provided a write up of their proposed problem and the collection has been edited by Klas Markstr\"om.Comment: This problem collection is published as part of the IML preprint series for the research programme and also available there http://www.mittag-leffler.se/research-programs/preprint-series?course_id=4401. arXiv admin note: text overlap with arXiv:1403.5975, arXiv:0706.4101 by other author

Spanning Cycles through Specified Edges in Bipartite Graphs

Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the number of edges in F, and let t1 and t2 be the numbers of components of F having odd and even length, respectively. We prove that G has a spanning cycle containing F if any two nonadjacent vertices in opposite partite sets have degree-sum at least n/2 + τ(F), where τ(F) = ⌈k/2 ⌉ + ɛ (here ɛ = 1 if t1 = 0 or if (t1, t2) ∈ {(1, 0), (2, 0)}, and ɛ = 0 otherwise). We show also that this threshold on the degree-sum is sharp when n> 3k.