On the chromatic roots of generalized theta graphs

The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)] k/\log k, uniformly in the path lengths s_i. Moreover, we prove that \Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph with a chromatic root that maximizes |z-1| is the one with all path lengths equal to 2; we conjecture that this holds for all k.Comment: LaTex2e, 25 pages including 2 figure

Coloring triangle-free rectangle overlap graphs with O(log⁡log⁡n)O(\log\log n) colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio

Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of $P(G,q)$ with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$ are presented and compared with series expansions and bounds.Comment: 33 pages, Latex, 5 postscript figures, published version; includes further comments on large-q serie

Algorithms for the minimum sum coloring problem: a review

The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex coloring problem which has a number of AI related applications. Due to its theoretical and practical relevance, MSCP attracts increasing attention. The only existing review on the problem dates back to 2004 and mainly covers the history of MSCP and theoretical developments on specific graphs. In recent years, the field has witnessed significant progresses on approximation algorithms and practical solution algorithms. The purpose of this review is to provide a comprehensive inspection of the most recent and representative MSCP algorithms. To be informative, we identify the general framework followed by practical solution algorithms and the key ingredients that make them successful. By classifying the main search strategies and putting forward the critical elements of the reviewed methods, we wish to encourage future development of more powerful methods and motivate new applications

Performance of distributed mechanisms for flow admission in wireless adhoc networks

Given a wireless network where some pairs of communication links interfere with each other, we study sufficient conditions for determining whether a given set of minimum bandwidth quality-of-service (QoS) requirements can be satisfied. We are especially interested in algorithms which have low communication overhead and low processing complexity. The interference in the network is modeled using a conflict graph whose vertices correspond to the communication links in the network. Two links are adjacent in this graph if and only if they interfere with each other due to being in the same vicinity and hence cannot be simultaneously active. The problem of scheduling the transmission of the various links is then essentially a fractional, weighted vertex coloring problem, for which upper bounds on the fractional chromatic number are sought using only localized information. We recall some distributed algorithms for this problem, and then assess their worst-case performance. Our results on this fundamental problem imply that for some well known classes of networks and interference models, the performance of these distributed algorithms is within a bounded factor away from that of an optimal, centralized algorithm. The performance bounds are simple expressions in terms of graph invariants. It is seen that the induced star number of a network plays an important role in the design and performance of such networks.Comment: 21 pages, submitted. Journal version of arXiv:0906.378

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of Proposition 4.1, and adds related discussion. To appear in Combinatorics, Probability & Computin