48 research outputs found
A Branch-and-Cut Strategy for the Manickam-Miklós-Singhi Conjecture
The Manickam-Miklós-Singhi Conjecture states that when n ≥ 4k, every multiset of n real numbers with nonnegative total sum has at least () n−1 k−1 k-subsets with nonnegative sum. We develop a branch-and-cut strategy using a linear programming formulation to show that verifying the conjecture for fixed values of k is a finite problem. To improve our search, we develop a zero-error randomized propagation algorithm. Using implementations of these algorithms, we verify a stronger form of the conjecture for all k ≤ 7
On the hardness of recognizing triangular line graphs
Given a graph G, its triangular line graph is the graph T(G) with vertex set
consisting of the edges of G and adjacencies between edges that are incident in
G as well as being within a common triangle. Graphs with a representation as
the triangular line graph of some graph G are triangular line graphs, which
have been studied under many names including anti-Gallai graphs, 2-in-3 graphs,
and link graphs. While closely related to line graphs, triangular line graphs
have been difficult to understand and characterize. Van Bang Le asked if
recognizing triangular line graphs has an efficient algorithm or is
computationally complex. We answer this question by proving that the complexity
of recognizing triangular line graphs is NP-complete via a reduction from
3-SAT.Comment: 18 pages, 8 figures, 4 table
Graph realizations constrained by skeleton graphs
In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM)
model to capture the fundamental difference in assortativity of networks in
nature studied by the physical and life sciences and social networks studied in
the social sciences. In 2014 Czabarka proposed a direct generalization of the
JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the
vertices have specified degrees, and the vertex set itself is partitioned into
classes. For each pair of vertex classes the number of edges between the
classes in a graph realization is prescribed. In this paper we apply the new
{\em skeleton graph} model to describe the same information as the PAM model.
Our model is more convenient for handling problems with low number of partition
classes or with special topological restrictions among the classes. We
investigate two particular cases in detail: (i) when there are only two vertex
classes and (ii) when the skeleton graph contains at most one cycle.Comment: 19 page
Minimal forbidden sets for degree sequence characterizations
Given a set F of graphs, a graph G is F-free if G does not contain any member of as an induced subgraph. A set F is degree-sequence-forcing (DSF) if, for each graph G in the class C of -free graphs, every realization of the degree sequence of G is also in C. A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality k. Using these properties and a computer search, we characterize the minimal DSF triples