scholarly journals Group coloring is Π2P-complete

2005 ◽  
Vol 349 (1) ◽  
pp. 99-111 ◽  
Author(s):  
Daniel Král’
Keyword(s):  
Group Coloring

scholarly journals Group Connectivity and Group Coloring: Small Groups versus Large Groups

10.37236/8984 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Rikke Langhede ◽  
Carsten Thomassen
Keyword(s):  
Abelian Group ◽  
Lower Bound ◽  
Small Groups ◽  
Upper Bound ◽  
Planar Graphs ◽  
General Concept ◽  
Group Connectivity ◽  
Group Coloring ◽  
Large Groups

A well-known result of Tutte says that if $\Gamma$ is an Abelian group and $G$ is a graph having a nowhere-zero $\Gamma$-flow, then $G$ has a nowhere-zero $\Gamma'$-flow for each Abelian group $\Gamma'$ whose order is at least the order of $\Gamma$. Jaeger, Linial, Payan, and Tarsi observed that this does not extend to their more general concept of group connectivity. Motivated by this we define $g(k)$ as the least number such that, if $G$ is $\Gamma$-connected for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-connected for every Abelian group $\Gamma'$ of order $|\Gamma'| \geqslant g(k)$. We prove that $g(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < g(k) \leqslant 8k^3+1.\end{align*} The upper bound holds for all $k$. Analogously, we define $h(k)$ as the least number such that, if $G$ is $\Gamma$-colorable for some Abelian group $\Gamma$ of order $k$, then $G$ is also $\Gamma'$-colorable for every Abelian group $\Gamma'$ of order $|\Gamma'| \geq h(k)$. Then $h(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < h(k) < (2+o(1))k \ln(k).\end{align*} The upper bound (for all $k$) follows from a result of Král', Pangrác, and Voss. The lower bound follows by duality from our lower bound on $g(k)$ as that bound is demonstrated by planar graphs.

scholarly journals Graph polynomials and group coloring of graphs

2022 ◽  
Vol 102 ◽  
pp. 103505
Author(s):  
Bartłomiej Bosek ◽  
Jarosław Grytczuk ◽  
Grzegorz Gutowski ◽  
Oriol Serra ◽  
Mariusz Zając
Keyword(s):  
Graph Polynomials ◽  
Group Coloring

Disproof of the group coloring version of the Hadwiger conjecture

2014 ◽  
Vol 06 (03) ◽  
pp. 1450035
Author(s):  
Baoyindureng Wu ◽  
Xinhui An
Keyword(s):  
Chromatic Number ◽  
Electronic Journal ◽  
Free Graph ◽  
Group Coloring

Inspired by the idea of Barát, Joret and Wood for disproving the list Hadwiger conjecture [The Electronic Journal of Combinatorics 18 (2011), P232], we disprove the group coloring version of the Hadwiger conjecture, which asserts that if G is a Kk-minor-free graph, then χg(G) ≤ k, where χg(G) denotes the group chromatic number of G.

HAJÓS-LIKE THEOREM FOR GROUP COLORING

2010 ◽  
Vol 02 (03) ◽  
pp. 433-436 ◽  
Author(s):  
XINHUI AN ◽  
BAOYINDURENG WU
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Chromatic Number ◽  
Graph Coloring ◽  
Complete Bipartite Graph ◽  
Graph Operations ◽  
Group Coloring

The group coloring of graphs is a new kind of graph coloring, introduced by Jaeger et al. in 1992, and the group chromatic number of a graph G is denoted by χg (G). In this note, we prove that for a positive integer k, a graph G with χg (G)>k can be obtained from any complete bipartite graph G0 with χg(G0)>k by certain types of graph operations.

scholarly journals A dual version of the Brooks group coloring theorem

2012 ◽  
Vol 312 (15) ◽  
pp. 2294-2303
Author(s):  
Deying Li ◽  
Suohai Fan ◽  
Hong-Jian Lai ◽  
Senmei Yao
Keyword(s):  
Dual Version ◽  
Group Coloring ◽  
Coloring Theorem

Group coloring via its geometric structures

2021 ◽  
Author(s):  
Muhammed Bello ◽  
Nor Muhainiah Mohd Ali
Keyword(s):  
Geometric Structures ◽  
Group Coloring
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