For r 4 we determine the smallest number of vertices, .<71-( d), of complete that are decomposable into two isomorphic factors for a given finite diameter d. We also prove that for a ,d such graph exists for each order than gr( d). 1. INTRODUCTORY NOTES AND DEFINITIONS In this paper we study decompositions of finite complete multipartite graphs factors with prescribed diameter. A factor F of a ...

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. In this paper, we obtain upper bounds for Castelnuovo-Mumford regularity of I(G)q in terms certain combinatorial invariants associated with G. We also prove weaker version conjecture by Alilooee, Banerjee, Beyarslan Hà on an bound conjectured class vertex decomposable graphs. Using these results, explicitly compute sev...

We study projective dimension, a graph parameter (denoted by pd(G) for a graph G), introduced by Pudlák and Rödl [13], who showed that proving lower bounds for pd(Gf ) for bipartite graphs Gf associated with a Boolean function f imply size lower bounds for branching programs computing f . Despite several attempts [13, 17], proving super-linear lower bounds for projective dimension of explicit f...

Block-intersection graphs of Steiner triple systems are considered. We prove that the block-intersection graphs of non-isomorphic Steiner triple systems are themselves non-isomorphic. We also prove that each Steiner triple system of order at most 15 has a Hamilton decomposable block-intersection graph.

Given a finite simple undirected graph G there is simplicial complex Ind(G), called the independence complex, whose faces correspond to independent sets of G. This well-studied concept because it provides fertile ground for interactions between commutative algebra, theory and algebraic topology. In this paper, we consider generalization complex. [Formula: see text], subset vertex set r-independ...

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, are assigned different colors. The strong chromatic index of G is the smallest number k for which G has a strong k-edge-coloring. A Halin graph is a planar graph consisting of a tree with no vertex of degree tw...

compiled April 30, 2009 from draft version hg:e0660c153c0b:79 An acyclic coloring of a graph is a proper vertex coloring such that the subgraph induced by the union of any two color classes is a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. The acyclic and star chromatic numbers ...

The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges graph respecting precedence constraints given by partial order on classes edges. We show that the special case with connected NP-hard even orders decomposable into chain and an incomparable class. For linearly ordered (possibly disconnected) classes, we get 5/3-approximations fixed-parameter algorithms transfe...

Let G=(V,E) be a graph, a subset X of V is an interval of G whenever for a, b E X and xE V X , (a,x)EE (resp. (x,a)EE) if and only if (b,x)EE (resp. (x,b)EE). For instance, 0, {x}, where x E V, and V are intervals of G, called trivial intervals. A graph G is then said to be indecomposable when all of its intervals are trivial. In the opposite case, we will say that G is decomposable. We now int...