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DTSTART;TZID=Asia/Seoul:20190520T163000
DTEND;TZID=Asia/Seoul:20190520T173000
DTSTAMP:20260423T105827
CREATED:20190304T123855Z
LAST-MODIFIED:20240707T090406Z
UID:642-1558369800-1558373400@dimag.ibs.re.kr
SUMMARY:Lars Jaffke\, A complexity dichotomy for critical values of the b-chromatic number of graphs
DESCRIPTION:A $b$-coloring of a graph $G$ is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The $b$-Coloring problem asks whether a graph $G$ has a $b$-coloring with $k$ colors.\nThe $b$-chromatic number of a graph $G$\, denoted by $\chi_b(G)$\, is the maximum number $k$ such that $G$ admits a $b$-coloring with $k$ colors. We consider the complexity of the $b$-Coloring problem\, whenever the value of $k$ is close to one of two upper bounds on $\chi_b(G)$: The maximum degree $\Delta(G)$ plus one\, and the $m$-degree\, denoted by $m(G)$\, which is defined as the maximum number $i$ such that $G$ has $i$ vertices of degree at least $i-1$. We obtain a dichotomy result stating that for fixed $k \in\{\Delta(G) + 1 − p\, m(G) − p\}$\, the problem is polynomial-time solvable whenever $p\in\{0\, 1\}$ and\, even when $k = 3$\, it is NP-complete whenever $p \ge 2$.\nWe furthermore consider parameterizations of the $b$-Coloring problem that involve the maximum degree $\Delta(G)$ of the input graph $G$ and give two FPT-algorithms. First\, we show that deciding whether a graph G has a $b$-coloring with $m(G)$ colors is FPT parameterized by $\Delta(G)$. Second\, we show that $b$-Coloring is FPT parameterized by $\Delta(G) + \ell_k(G)$\, where $\ell_k(G)$ denotes the number of vertices of degree at least $k$.\nThis is joint work with Paloma T. Lima.
URL:https://dimag.ibs.re.kr/event/2019-05-20/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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