Ear Decomposition and Balanced Neighborly Simplicial Manifolds

We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.

Regularity of the vanishing ideal over a bipartite nested ear decomposition

We study the Castelnuovo–Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length [Formula: see text] increases by [Formula: see text], where [Formula: see text] is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, [Formula: see text], endowed with a weak nested ear decomposition is equal to [Formula: see text] where [Formula: see text] is the number of even length ears and pendant edges of the decomposition. As a corollary, we show that for bipartite graph the number of even length ears in a nested ear decomposition starting from a vertex is constant.

Matching Covered Graphs with Three Removable Classes

The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. The last (single or double) ear of an ear decomposition is defined as a removable class. Every matching covered graph not induced by a circuit has at least three removable classes. In this paper, we characterize matching covered graphs with precisely three removable classes and show, as a corollary, that every non-planar matching covered graph has at least four removable classes. Let $G$ be a matching covered graph. A matching covered subgraph $H$ of $G$ is conformal if $G-VH$ has a perfect matching. Given $S \subseteq EG$, what is a minimal conformal subgraph of $G$ that contains $S$? It is known that if $|S|=2$ then it is induced by a circuit. As an application of the main result, we answer this question for $|S|=3$.

ON FINDING SPARSE THREE-EDGE-CONNECTED AND THREE-VERTEX-CONNECTED SPANNING SUBGRAPHS

We present algorithms that construct a sparse spanning subgraph of a three-edge-connected graph that preserves three-edge connectivity or of a three-vertex-connected graph that preserves three-vertex connectivity. Our algorithms are conceptually simple and run in O(|E|) time. These simple algorithms can be used to improve the efficiency of the best-known algorithms for three-edge and three-vertex connectivity and their related problems, by preprocessing the input graph so as to trim it down to a sparse graph. Afterwards, the original algorithms run in O(|V|) instead of O(|E|) time. Our algorithms generate an adjacency-lists structure to represent the sparse spanning subgraph, so that when a depth-first search is performed over the subgraph based on this adjacency-lists structure it actually traverses the paths in an ear-decomposition of the subgraph. This is useful because many of the existing algorithms for three-edge- or three-vertex connectivity and their related problems are based on an ear-decomposition of the given graph. Using such an adjacency-lists structure to represent the input graph would greatly improve the run-time efficiency of these algorithms.