T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions

We present exact calculations of the zero-temperature partition function for the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width $L_y=4$ and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e. of torus or Klein bottle type). These boundary conditions have the advantage of removing edge effects. In the limit of infinite length, we calculate the exponent of the entropy, $W(q)$ and determine the continuous locus ${\cal B}$ where it is singular. We also give results for toroidal strips involving ``crossing subgraphs''; these make possible a unified treatment of torus and Klein bottle boundary conditions and enable us to prove that for a given strip, the locus ${\cal B}$ is the same for these boundary conditions.Comment: 43 pages, latex, 4 postscript figure

Total coloring of 1-toroidal graphs of maximum degree at least 11 and no adjacent triangles

A {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by \chiup''(G), is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $\Delta$ admits a total coloring with at most $\Delta + 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of $1$-toroidal graphs, and prove that the TCC holds for the $1$-toroidal graphs with maximum degree at least~$11$ and some restrictions on the triangles. Consequently, if $G$ is a $1$-toroidal graph with maximum degree $\Delta$ at least~$11$ and without adjacent triangles, then $G$ admits a total coloring with at most $\Delta + 2$ colors.Comment: 10 page

On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

Experimental Evaluation of Book Drawing Algorithms

A $k$-page book drawing of a graph $G=(V,E)$ consists of a linear ordering of its vertices along a spine and an assignment of each edge to one of the $k$ pages, which are half-planes bounded by the spine. In a book drawing, two edges cross if and only if they are assigned to the same page and their vertices alternate along the spine. Crossing minimization in a $k$-page book drawing is NP-hard, yet book drawings have multiple applications in visualization and beyond. Therefore several heuristic book drawing algorithms exist, but there is no broader comparative study on their relative performance. In this paper, we propose a comprehensive benchmark set of challenging graph classes for book drawing algorithms and provide an extensive experimental study of the performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017