On Choosability and Paintability of Graphs

abstract: Let $G=(V,E)$ be a graph. A \emph{list assignment} $L$ for $G$ is a function from $V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$ with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$ whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$ there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al. suggested the problem of determining \ensuremath{\ch(K_{s*k})}, and showed that $\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that $\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$. An online version of the list coloring problem was introduced independently by Schauz and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice designs lists of colors for all vertices, but does not tell Bob, and is allowed to change her mind about unrevealed colors as the game progresses. On her $i$-th turn Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides, irrevocably, which (independent set) of these vertices to color with $i$. For a function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If $l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable. The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$ is $t$-paintable, and is denoted by \ensuremath{\ch^{\mathrm{OL}}(G)}. Evidently, $\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$ can be arbitrarily large. However there are only a few known examples with this gap equal to one, and none with larger gap. This conjecture is explored in this thesis. One of the obstacles is that there are not many graphs whose exact list coloring number is known. This is one of the motivations for establishing new cases of Erd\H{o}s' problem. Here new examples of graphs with gap one are found, and related technical results are developed as tools for attacking Zhu's conjecture. The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$. This was recently disproved for specially constructed graphs. Here it is shown that a graph arising naturally in the theory of cellular networks is also a counterexample.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Radio Co-location Aware Channel Assignments for Interference Mitigation in Wireless Mesh Networks

Designing high performance channel assignment schemes to harness the potential of multi-radio multi-channel deployments in wireless mesh networks (WMNs) is an active research domain. A pragmatic channel assignment approach strives to maximize network capacity by restraining the endemic interference and mitigating its adverse impact on network performance. Interference prevalent in WMNs is multi-faceted, radio co-location interference (RCI) being a crucial aspect that is seldom addressed in research endeavors. In this effort, we propose a set of intelligent channel assignment algorithms, which focus primarily on alleviating the RCI. These graph theoretic schemes are structurally inspired by the spatio-statistical characteristics of interference. We present the theoretical design foundations for each of the proposed algorithms, and demonstrate their potential to significantly enhance network capacity in comparison to some well-known existing schemes. We also demonstrate the adverse impact of radio co- location interference on the network, and the efficacy of the proposed schemes in successfully mitigating it. The experimental results to validate the proposed theoretical notions were obtained by running an exhaustive set of ns-3 simulations in IEEE 802.11g/n environments.Comment: Accepted @ ICACCI-201

A Coloring Algorithm for Disambiguating Graph and Map Drawings

Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down eye movements. In this paper we propose an algorithm that disambiguates the edges with automatic selection of distinctive colors. Our proposed algorithm computes a near optimal color assignment of a dual collision graph, using a novel branch-and-bound procedure applied to a space decomposition of the color gamut. We give examples demonstrating the effectiveness of this approach in clarifying drawings of real world graphs and maps