A Polynomial Method for Counting Colorings of SS-labeled Graphs

The notion of $S$-labeling, where $S$ is a subset of the symmetric group, is a common generalization of signed $k$-coloring, signed $\mathbb{Z}_k$-coloring, DP-coloring, group coloring, and coloring of gained graphs that was introduced in 2019 by Jin, Wong, and Zhu. In this paper, we present a unified and simple polynomial method for giving exponential lower bounds on the number of colorings of an $S$-labeled graph. This algebraic technique allows us to prove new lower bounds on the number of colorings of any $S$-labeling of graphs satisfying certain sparsity conditions. This gives new lower bounds on the DP color function, and consequently chromatic polynomial and list color function, of families of planar graphs, and the number of colorings of signed graphs. These bounds improve previously known results, or are the first such known results.Comment: 25 pages, 2 figure

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter $\alpha\in\mathbb{N}$. Nevertheless, here it is proved that the probability of requiring a value of $\alpha>k$ to obtain a solution for a random graph decreases exponentially: $P(\alpha>k) \leq 2^{-(k+1)}$, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

Graph Polynomials and Group Coloring of Graphs

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the elements of $\Gamma$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.Comment: 14 page

Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles

We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D\u3e0$ such that if $H$ is a planar triangle-free graph and if $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a 3-coloring of $H$. This gives a positive answer to a conjecture of Dvořák, Král\u27, and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree

Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page