A k-cycle decomposition of a multipartite graph G is said to be gregarious if each k-cycle in the decomposition intersects k distinct partite sets of G. In this paper we prove necessary and sufficient conditions for the existence of such a decomposition in the case where G is the complete equipartite graph, having n parts of size m, and either n ≡ 0, 1 (mod k), or k is odd and m ≡ 0 (mod k). As...

An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-c...

We consider the dihamiltonian decomposition problem for 3-regular graphs. A graph G is dihamiltonian decomposable if in the digraph obtained from G by replacing each edge of G as two directed edges, the set of edges are partitioned into 3 edge-disjoint directed hamiltonian cycles. We suggest some conditions for dihamiltonian decomposition of 3-regular graphs: for a 3-regular graph G, it is diha...

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with ( n+1 2 ) edges admits an edge decomposition G = H1 ⊕ · · · ⊕ Hn such that Hi has i edges and it is isomorphic to a subgraph of Hi+1, i = 1, . . . , n − 1. We show that every bipartite graph G with ( n+1 2 ) edges such that the degree sequence d1, . . . , dk of one of the stable sets satisfies di ≥ n − i + 1, ...

A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n+mlogn) algorithm that uses this duality to find both a trans...

A connected graph G is said to be factor-critical if G − v has a perfect matching for every vertex v of G. Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs G satisfying that G− v has a unique perfect matching for any vertex v of G with degree at least 3 is characterized. From this, the number of maximum m...

In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product...

We prove that a graph has an r-bounded subdivision of wheel if and only it does not have graph-decomposition locality r width at most two.

In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node; then, the number of spanning trees of G is computed as the product...

To solve task decomposition of multi-expert system, agent-based task decomposition model and algorithm based on dependency is introduced. Firstly, AND/OR dependency graph for operation set is presented. Formal description of task decomposition based on With AND/OR dependency graph, is proposed. Secondly, task decomposition algorithm based on operation dependency is analyzed to solve communicati...