Approximately Coloring Graphs Without Long Induced Paths
نویسندگان
چکیده
منابع مشابه
Approximately Coloring Graphs Without Long Induced Paths
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on t vertices, for fixed t. We propose an algorithm that, given a 3-colorable graph without an induced path on t vertices, computes a coloring with max { 5, 2 ⌈ t−1 2 ⌉ − 2 } many colors. If the input graph is triangle-free, we only need max { 4, ⌈ t−1...
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Let Pk be a path, Ck a cycle on k vertices, and Kk,k a complete bipartite graph with k vertices on each side of the bipartition. We prove that (1) for any integers k, t > 0 and a graph H there are finitely many subgraph minimal graphs with no induced Pk and Kt,t that are not Hcolorable and (2) for any integer k > 4 there are finitely many subgraph minimal graphs with no induced Pk that are not ...
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We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a P 8-free graph is 5-colorable and of deciding whether a P 12-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a P 5-free graph is 3-colorable.
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For an integer k ≥ 1, a graph G is k-colorable if there exists a mapping c : VG → {1, . . . , k} such that c(u) 6= c(v) whenever u and v are two adjacent vertices. For a fixed integer k ≥ 1, the k-COLORING problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g ≥ 4 we determine a lower bound `(g), such that ...
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We show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an NP-complete problem, improving a previous NP-completeness result proved by Woeginger and Sgall in 2001. The complexity of 4-coloring graphs without induced paths on five vertices remains open. We show that deciding if a graph without induced paths or cycles on five vertices can be colored ...
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ژورنال
عنوان ژورنال: Algorithmica
سال: 2019
ISSN: 0178-4617,1432-0541
DOI: 10.1007/s00453-019-00577-6