Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k-cycles, Australas. J. Combin. 18 (1998) 193-200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m, has an edge-disjoint decomposition into paths of length k if and only if k divides 3m² and k<3m. Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size m has an edge-disjoint decomposition into cycles (and also into paths) of length k with k?3 if and only if k divides 10m² and k?5m for cycles (or k<5m for paths).     

Equipartite gregarious 6- and 8-cycle systems

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite gregarious 3-cycle systems are 3-GDDs, and necessary and sufficient conditions for their existence are known (see for instance the CRC Handbook of Combinatorial Designs, 1996, C.J. Colbourn, J.H. Dinitz (Eds.), Section III 1.3). The cases of equipartite and of almost equipartite 4-cycle systems were recently dealt with by Billington and Hoffman. Here, for both 6-cycles and for 8-cycles, we give necessary and sufficient conditions for existence of a gregarious cycle decomposition of the complete equipartite graph K_n(a) (with n parts, n?6 or n?8, of size a).     

Equipartite Gregarious 5-Cycle Systems and Other Results

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph K_m(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).     

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.     

Decomposing Complete Equipartite Graphs into Closed Trails of Length k

Necessary conditions for a simple connected graph G to admit a decomposition into closed trails of length k ≥ 3 are that G is even and its total number of edges is a multiple of k. In this paper we show that these conditions are sufficient in the case when G is the complete equipartite graph having at least three parts, each of the same size.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

An extremal problem in hypergraph coloring

F(n, r, k) is defined to be the minimum number of edges in an r graph on n vertices in which there exists a strongly colored edge in every equipartite k-coloring. Approximate values are given for this function in the general case, and the problem is solved in the particular case of graphs.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Minimum graphs with complete k-closure

The k-closure of a graph G, as defined by Bondy and Chvátal, is the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree-sum at least k. We determine the number of edges necessary for the k-closure of a graph to be complete.     

Bounds on the eigenvalues of graphs with cut vertices or edges

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.     

A new lower bound on the number of trivially noncontractible edges in contraction critical 5-connected graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. An edge of a k-connected graph is called trivially noncontractible if its two end vertices have a common neighbor of degree k. Ando [K. Ando, Trivially noncontractible edges in a contraction critically 5-connected graph, Discrete Math. 293 (2005) 61-72] proved that a contraction critical 5-connected graph on n vertices has at least n/2 trivially noncontractible edges. Li [Xiangjun Li, Some results about the contractible edge and the domination number of graphs, Guilin, Guangxi Normal University, 2006 (in Chinese)] improved the lower bound to n+1. In this paper, the bound is improved to the statement that any contraction critical 5-connected graph on n vertices has at least trivially noncontractible edges.     

Paths with a given number of vertices from each partite set in regular multipartite tournaments

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph.For c?2 we prove that a regular c-partite tournament with r?2 vertices in each partite set contains a directed path with exactly two vertices from each partite set. Furthermore, if c?4, then we will show that almost all regular c-partite tournaments D contain a directed path with exactly r-s vertices from each partite set for each given integer s∈N, if r is the cardinality of each partite set of D. Some related results are also presented.     

The competition number of a graph whose holes do not overlap much

Let D be an acyclic digraph. The competition graph of D is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x in D such that (u,x) and (v,x) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of G is the smallest number of such isolated vertices.A hole of a graph is an induced cycle of length at least four. Kim (2005) [8] conjectured that the competition number of a graph with h holes is at most h+1. Recently, Li and Chang (2009) [11] showed that the conjecture is true when the holes are independent. In this paper, we show that the conjecture is true though the holes are not independent but mutually edge-disjoint.     

Resolvable path designs

A resolvable (balanced) path design, RBPD(v, k, λ) is the decomposition of λ copies of the complete graph on v vertices into edge-disjoint subgraphs such that each subgraph consists of $\text{v}\text{k}$ vertex-disjoint paths of length k ? 1 (k vertices). It is shown that an RBPD(v, 3, λ) exists if and only if v ≡ 9 (modulo 12/gcd(4, λ)). Moreover, the RBPD(v, 3, λ) can have an automorphism of order $\text{v}\text{3}$. For k > 3, it is shown that if v is large enough, then an RBPD(v, k, 1) exists if and only if v ≡ k² (modulo lcm(2k ? 2, k)). Also, it is shown that the categorical product of a k-factorable graph and a regular graph is also k-factorable. These results are stronger than two conjectures of P. Hell and A. Rosa     

Some remarks on the odd hadwiger’s conjecture

We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l ? 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K _k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K _k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) ? 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.     

Closed trail decompositions of complete equipartite graphs

The complete equipartite graph $K_m * {\overline{K_n}}$ has mn vertices partitioned into m parts of size n, with two vertices adjacent if and only if they are in different parts. In this paper, we determine necessary and sufficient conditions for the existence of a decomposition of $K_m * {\overline{K_n}}$ into closed trails of length k. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 374–403, 2009     

Upper Bound on the Diameter of a Domination Dot-Critical Graph

A vertex of a graph is called critical if its deletion decreases the domination number, and an edge is called dot-critical if its contraction decreases the domination number. A graph is said to be dot-critical if all of its edges are dot-critical. In this paper, we show that if G is a connected dot-critical graph with domination number k??? 3 and diameter d and if G has no critical vertices, then d??? 2k?3.     

A Note on Large Rainbow Matchings in Edge-coloured Graphs

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided ${n\geq \frac{17}{4}k^2}$ . In this paper, we show that n ≥ 4k ? 4 is sufficient for k ≥ 4.     

Bounds on the number of disjoint spanning trees

It is shown that an n-edge connected graph has at least ?$\text{(n ? 1)}\text{2}$? pairwise edge-disjoint spanning trees. This bound is best possible in general. A maximal planar graph with four or more vertices contains two edge-disjoint spanning trees. For a maximal toroidal graph, this number is three.