scholarly journals Colorings and Orientations of Matrices and Graphs

10.37236/1087 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Planar Graphs ◽  
Partial Solution ◽  
General Concept ◽  
List Coloring ◽  
Weighted Sum ◽  
Incidence Matrices ◽  
Graph Theoretic ◽  
Four Color Theorem ◽  
List Colorings

We introduce colorings and orientations of matrices as generalizations of the graph theoretic terms. The permanent per$(A[\zeta|\xi])$ of certain copies $A[\zeta|\xi]$ of a matrix $A$ can be expressed as a weighted sum over the orientations or the colorings of $A$. When applied to incidence matrices of graphs these equations include Alon and Tarsi's theorem about Eulerian orientations and the existence of list colorings. In the case of planar graphs we deduce Ellingham and Goddyn's partial solution of the list coloring conjecture and Scheim's equivalency between not vanishing permanents and the four color theorem. The general concept of matrix colorings in the background is also connected to hypergraph colorings and matrix choosability.

scholarly journals Distributed Coloring in Sparse Graphs with Fewer Colors

10.37236/8395 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pierre Aboulker ◽  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Louis Esperet
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Distributed Algorithm ◽  
Planar Graphs ◽  
List Coloring ◽  
Sparse Graphs ◽  
Four Color Theorem

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\text{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.

scholarly journals On $t$-Common List-Colorings

10.37236/6738 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Hojin Choi ◽  
Young Soo Kwon
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Four Color Theorem ◽  
List Colorings ◽  
List Chromatic Number ◽  
Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

scholarly journals A Note on Not-4-List Colorable Planar Graphs

10.37236/7320 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Margit Voigt ◽  
Arnfried Kemnitz
Keyword(s):  
Planar Graph ◽  
Bipartite Graph ◽  
Planar Graphs ◽  
The Other ◽  
Four Color Theorem ◽  
List Colorings ◽  
Vertex Set ◽  
Colorable Graph ◽  
Negative Strengthening

The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-$4$-list colorable. In this paper we investigate a problem combining proper colorings and list colorings. We ask whether the vertex set of every planar graph can be partitioned into two subsets where one subset induces a bipartite graph and the other subset induces a $2$-list colorable graph. We answer this question in the negative strengthening the result on non-$4$-list colorable planar graphs.

scholarly journals 3-list-coloring planar graphs of girth 4

2011 ◽  
Vol 311 (6) ◽  
pp. 413-417 ◽  
Author(s):  
Jun-Lin Guo ◽  
Yue-Li Wang
Keyword(s):  
Planar Graphs ◽  
List Coloring

List coloring triangle‐free planar graphs

2020 ◽  
Vol 94 (2) ◽  
pp. 278-298
Author(s):  
Jianzhang Hu ◽  
Xuding Zhu
Keyword(s):  
Planar Graphs ◽  
List Coloring

scholarly journals 5-list-coloring planar graphs with distant precolored vertices

2017 ◽  
Vol 122 ◽  
pp. 311-352 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Bernard Lidický ◽  
Bojan Mohar ◽  
Luke Postle
Keyword(s):  
Planar Graphs ◽  
List Coloring

Lifted Inference with Tree Axioms

2021 ◽  
Author(s):  
Timothy van Bremen ◽  
Ondřej Kuželka
Keyword(s):  
Binary Relation ◽  
Domain Size ◽  
Weighted Sum ◽  
Order Model ◽  
Lifted Inference ◽  
Past Work ◽  
First Order ◽  
Graph Theoretic ◽  
Model Counting ◽  
The Given

We consider the problem of weighted first-order model counting (WFOMC): given a first-order sentence ϕ and domain size n ∈ ℕ, determine the weighted sum of models of ϕ over the domain {1, ..., n}. Past work has shown that any sentence using at most two logical variables admits an algorithm for WFOMC that runs in time polynomial in the given domain size (Van den Broeck 2011; Van den Broeck, Meert, and Darwiche 2014). In this paper, we extend this result to any two-variable sentence ϕ with the addition of a tree axiom, stating that some distinguished binary relation in ϕ forms a tree in the graph-theoretic sense.

scholarly journals Equitable list colorings of planar graphs without short cycles

2008 ◽  
Vol 407 (1-3) ◽  
pp. 21-28 ◽  
Author(s):  
Junlei Zhu ◽  
Yuehua Bu
Keyword(s):  
Planar Graphs ◽  
List Colorings
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