Complexity of conditional colorability of graphs


Li, Xueliang and Yao, Xiangmei and Zhou, Wenli and Broersma, Hajo (2009) Complexity of conditional colorability of graphs. Applied Mathematics Letters, 22 (3). pp. 320-324. ISSN 0893-9659

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Abstract:For positive integers $k$ and $r$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $\min\{r,d(v)\}$ different colors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th-order conditional chromatic number, and is denoted by $X_r(G)$. It is easy to see that conditional coloring is a generalization of traditional vertex coloring (the case $r=1$). In this work, we consider the complexity of the conditional colorability of graphs. Our main result is that conditional (3,2)-colorability remains NP-complete when restricted to planar bipartite graphs with maximum degree at most 3 and arbitrarily high girth. This differs considerably from the well-known result that traditional 3-colorability is polynomially solvable for graphs with maximum degree at most 3. On the other hand we show that (3,2)-colorability is polynomially solvable for graphs with bounded tree-width. We also prove that some other well-known complexity results for traditional coloring still hold for conditional coloring.

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Copyright:© 2009 Elsevier
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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