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 prove that all connected vertex-transitive graphs of order p, p a prime, can be decomposed into Hamilton cycles.

Given any finite simplicial complex ∆, we show how to construct from a colouring χ of ∆ a new simplicial complex ∆χ that is balanced and vertex decomposable. In addition, the h-vector of ∆χ is precisely the f -vector of ∆. Our construction generalizes the “whiskering” construction of Villarreal, and Cook and Nagel. We also reverse this construction to prove a special case of a conjecture of Coo...

The graph G(N; d) has vertex set V = {0; 1; : : : ; N − 1}, with {v; w} an edge if v − w ≡ ±di(modN ) for some 06i6dlogd Ne − 1. We show that the circulant graph G(cd; d) is Hamilton decomposable for all positive integers c; d, and m with c¡d. This extends work of Micheneau and answers a special case of a question of Alspach. c © 2000 Elsevier Science B.V. All rights reserved.

We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn, n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there...

This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomp...

This thesis deals with the multi-terminal vertex separator problem. Given a graph G = (V ∪T,E) with V ∪T the set of vertices, where T is a set of terminals, and a weight function w : V → Z, associated with nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V ∪T into k+1 subsets {S, V1, . . . , Vk} such that there is no edge between two different subsets Vi a...

We construct the first explicit example of a simplicial 3-ball B15,66 that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B12,38 with 12 vertices that is collapsible and not shellable, but evasive. Finally, we present the first explicit triangulation of a 3-sphere S18,125 (with only 18 vertices) that is not locally constructible. All these examples are based on knotted ...