Whiskers and sequentially Cohen–Macaulay graphs

Whiskers and sequentially Cohen-Macaulay graphs

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially CohenMacaulay graph. We focus on modifications given by adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex in G. We give various sufficient conditions and...

SEQUENTIALLY Sr SIMPLICIAL COMPLEXES AND SEQUENTIALLY S2 GRAPHS

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre’s condition Sr . In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton...

Sequentially Cohen-Macaulay Graphs of Form θ_{n1, ...nk}

Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we cla...

Further Results on Sequentially Additive Graphs

Given a graph G with p vertices, q edges and a positive integer k, a k-sequentially additive labeling of G is an assignment of distinct numbers k, k + 1, k + 2, . . . , k + p + q − 1 to the p + q elements of G so that every edge uv of G receives the sum of the numbers assigned to the vertices u and v. A graph which admits such an assignment to its elements is called a k-sequentially additive gr...