Group Coloring and List Group Coloring Are Π2 P -Complete

Group Connectivity and Group Coloring: Small Groups versus Large Groups

The Electronic Journal of Combinatorics ◽

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.

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A max-group coloring-based gossiping mechanism using double-face phased array radar for wireless multi-hop networks

2017 First International Conference on Electronics Instrumentation & Information Systems (EIIS) ◽

10.1109/eiis.2017.8298550 ◽

2017 ◽

Author(s):

Ningxin Liu ◽

Laixian Peng ◽

Renhui Xu ◽

Jin Zhang ◽

Qi Wang ◽

...

Keyword(s):

Phased Array ◽

Phased Array Radar ◽

Group Coloring

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Graph polynomials and group coloring of graphs

European Journal of Combinatorics ◽

10.1016/j.ejc.2021.103505 ◽

2022 ◽

Vol 102 ◽

pp. 103505

Author(s):

Bartłomiej Bosek ◽

Jarosław Grytczuk ◽

Grzegorz Gutowski ◽

Oriol Serra ◽

Mariusz Zając

Keyword(s):

Graph Polynomials ◽

Group Coloring

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Disproof of the group coloring version of the Hadwiger conjecture

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500359 ◽

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.

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HAJÓS-LIKE THEOREM FOR GROUP COLORING

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830910000772 ◽

2010 ◽

Vol 02 (03) ◽

pp. 433-436 ◽

Cited By ~ 1

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.

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