# List-coloring the square of a subcubic graph

@article{Cranston2008ListcoloringTS,
title={List-coloring the square of a subcubic graph},
author={Daniel W. Cranston and Seog-Jin Kim},
journal={J. Graph Theory},
year={2008},
volume={57},
pages={65-87}
}
• Published 28 February 2015
• Mathematics, Computer Science
• J. Graph Theory
The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$ satisfies $\chi(G^2)\leq 7$. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$, that is $\chi_l(G^2)=\chi(G^2)$ for all $G$. If true, this conjecture (together with Thomassen's result…
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Coloring squares of planar graphs with no short cycles
• Mathematics
• 2005
Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the chromatic number of the square of every planar graph of girth at least g and maximum degree ∆ ≥ M(g) is ∆ + 1.
A bound on the chromatic number of the square of a planar graph
• Computer Science, Mathematics
J. Comb. Theory, Ser. B
• 2005
Choosability conjectures and multicircuits
• Mathematics, Computer Science
Discret. Math.
• 2001
On Moore Graphs with Diameters 2 and 3
• Mathematics, Computer Science
IBM J. Res. Dev.
• 1960
The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph to prove the existence of connected, undirected graphs homogeneous of degree d and of diameter k.
Introduction to Graph Theory
1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration.
List Colouring Squares of Planar Graphs
• Mathematics, Computer Science
Electron. Notes Discret. Math.
• 2007
A solution to a colouring problem of P. Erdös
• Computer Science, Mathematics
Discret. Math.
• 1992