TY - JOUR

T1 - λ-backbone colorings along pairwise disjoint stars and matchings

AU - Broersma, Haitze J.

AU - Fujisawa, J.

AU - Marchal, L.

AU - Paulusma, Daniël

AU - Salman, M.

AU - Yoshimoto, K.

PY - 2009/9/28

Y1 - 2009/9/28

N2 - Given an integer $\lambda \ge 2$, a graph $G=(V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\lambda$-backbone coloring of $(G,H)$ is a proper vertex coloring $V\to\{1,2,\ldots\}$ of $G$, in which the colors assigned to adjacent vertices in $H$ differ by at least $\lambda$. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone $S$ of $G$ the minimum number $\ell$ for which a $\lambda$-backbone coloring of $(G,S)$ with colors in $\{1,\ldots,\ell\}$ exists can roughly differ by a multiplicative factor of at most $2-{1\over \lambda}$ from the chromatic number $\chi(G)$. For the special case of matching backbones this factor is roughly $2-{2\over\lambda +1}$. We also show that the computational complexity of the problem “Given a graph $G$ with a star backbone $S$, and an integer $\ell$, is there a $\lambda$-backbone coloring of $(G,S)$ with colors in $\{1,\ldots,\ell\}$?��? jumps from polynomially solvable to NP-complete between $\ell=\lambda+1$ and $\ell =\lambda+2$ (the case $\ell=\lambda+2$ is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

AB - Given an integer $\lambda \ge 2$, a graph $G=(V,E)$ and a spanning subgraph $H$ of $G$ (the backbone of $G$), a $\lambda$-backbone coloring of $(G,H)$ is a proper vertex coloring $V\to\{1,2,\ldots\}$ of $G$, in which the colors assigned to adjacent vertices in $H$ differ by at least $\lambda$. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone $S$ of $G$ the minimum number $\ell$ for which a $\lambda$-backbone coloring of $(G,S)$ with colors in $\{1,\ldots,\ell\}$ exists can roughly differ by a multiplicative factor of at most $2-{1\over \lambda}$ from the chromatic number $\chi(G)$. For the special case of matching backbones this factor is roughly $2-{2\over\lambda +1}$. We also show that the computational complexity of the problem “Given a graph $G$ with a star backbone $S$, and an integer $\ell$, is there a $\lambda$-backbone coloring of $(G,S)$ with colors in $\{1,\ldots,\ell\}$?��? jumps from polynomially solvable to NP-complete between $\ell=\lambda+1$ and $\ell =\lambda+2$ (the case $\ell=\lambda+2$ is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

U2 - 10.1016/j.disc.2008.04.007

DO - 10.1016/j.disc.2008.04.007

M3 - Article

VL - 309

SP - 5596

EP - 5609

JO - Discrete mathematics

JF - Discrete mathematics

SN - 0012-365X

IS - 18

ER -