Magic graphs and the faces of the Birkhoff polytope

Magic labelings of graphs are studied in great detail by Stanley and Stewart. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.Comment: 9 page

The eigen-chromatic ratio of classes of graphs : molecular stability, asymptotes and area.

Master of Science in Mathematics, University of KwaZulu-Natal, Westville, 2017.This dissertation involves combining the two concepts of energy and the chromatic number of classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this ratio is the importance of its asymptotic convergence in applications, as well as the idea of area involving the Rieman integral of this ratio, when it is a function of the order n of the graph G belonging to a class of graphs. The energy of a graph G, is the sum of the absolute values of the eigenvalues associated with the adjacency matrix of G, and its importance has found its way into many areas of research in graph theory. The chromatic number of a graph G, is the least number of colours required to colour the vertices of the graph, so that no two adjacent vertices receive the same colour. The importance of ratios in graph theory is evident by the vast amount of research articles: Expanders, The central ratio of a graph, Eigen-pair ratio of classes of graphs , Independence and Hall ratios, Tree-cover ratio of graphs, Eigen-energy formation ratio, The eigen-complete difference ratio, The chromatic-cover ratio and "Graph theory and calculus: ratios of classes of graphs". We combine the two concepts of energy and chromatic number (which involves the order n of the graph G) in a ratio, called the eigen-chromatic ratio of a graph. The chromatic number associated with the molecular graph (the atoms are vertices and edges are bonds between the atoms) would involve the partitioning of the atoms into the smallest number of sets of like atoms so that like atoms are not bonded. This ratio would allow for the investigation of the effect of the energy on the atomic partition, when a large number of atoms are involved. The complete graph is associated with the value 1 2 when the eigen-chromatic ratio is investigated when a large number of atoms are involved; this has allowed for the investigation of molecular stability associated with the idea of hypo/hyper energetic graphs. Attaching the average degree to the Riemann integral of this ratio (as a function of n) would result in an area analogue for investigation. Once the ratio is defned the objective is to find the eigen-chromatic ratio of various well known classes of graphs such as the complete graph, bipartite graphs, star graphs with rays of length two, wheels, paths, cycles, dual star graphs, lollipop graphs and caterpillar graphs. Once the ratio of each class of graph are determined the asymptote and area of this ratio are determined and conclusions and conjectures inferred

On Box-Perfect Graphs

Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We also develop a general and powerful method for establishing box-perfectness

TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite

The main result of this work is that an orthogonal access scheme such as TDMA achieves the all-unicast degrees of freedom (DoF) region of the topological interference management (TIM) problem if and only if the network topology graph is chordal bipartite, i.e., every cycle that can contain a chord, does contain a chord. The all-unicast DoF region includes the DoF region for any arbitrary choice of a unicast message set, so e.g., the results of Maleki and Jafar on the optimality of orthogonal access for the sum-DoF of one-dimensional convex networks are recovered as a special case. The result is also established for the corresponding topological representation of the index coding problem

Wide partitions, Latin tableaux, and Rota's basis conjecture

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of mu). Then Brian Taylor and the first author have conjectured that an integer partition lambda is wide if and only if there exists a tableau of shape lambda such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy ``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results; 28 pages, submitted to Adv. Appl. Mat