List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraphs

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-03-20 DOI:10.1137/22m1515768

Sepehr Hajebi, Yanjia Li, Sophie Spirkl

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1158-1190, March 2024.
Abstract. The List-3-Coloring Problem is to decide, given a graph [math] and a list [math] of colors assigned to each vertex [math] of [math], whether [math] admits a proper coloring [math] with [math] for every vertex [math] of [math], and the 3-Coloring Problem is the List-3-Coloring Problem on instances with [math] for every vertex [math] of [math]. The List-3-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to [math]-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph [math]), it remains NP-complete unless [math] is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the 3-Coloring Problem on [math]-free graphs (where [math] denotes the eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs [math] and [math], we say [math] is [math]-free if [math] is not isomorphic to an induced subgraph of [math] with the isomorphism preserving the linear order. We prove, assuming [math] to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to [math]-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if [math] has at most one edge, and remains NP-complete if [math] has at least three edges. Moreover, in the case where [math] has exactly two edges, we give a complete dichotomy when the two edges of [math] share an end, and prove several NP-completeness results when the two edges of [math] do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.

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列表-3-使用禁止诱导子图为有序图着色

SIAM 离散数学杂志》，第 38 卷第 1 期，第 1158-1190 页，2024 年 3 月。摘要。列表-3-着色问题（List-3-Coloring Problem）是给定一个图[math]和一个分配给[math]的每个顶点[math]的颜色列表[math]，判断[math]是否允许一个适当的着色[math]，[math]的每个顶点[math]都有[math]，3-着色问题就是[math]的每个顶点[math]都有[math]的实例上的列表-3-着色问题。众所周知，List-3-Coloring Problem 是一个经典的 NP-complete（NP-完全）问题，虽然它仅限于无[math]图（指没有与固定图[math]同构的诱导子图），但除非[math]与路径的诱导子图同构，否则它仍然是 NP-完全的。然而，目前的技术水平还远未证明这足以实现多项式时间算法；事实上，无[math]图（其中[math]表示八顶点路径）上的 3-Coloring 问题的复杂度尚不可知。在这里，我们考虑的是列表-3-着色问题的一个变体，称为有序图列表-3-着色问题，其输入是有序图，即顶点集具有线性顺序的图。对于有序图[math]和[math]，如果[math]与[math]的诱导子图不同构，且同构时保留了线性顺序，我们就说[math]是无[math]的。假设 [math] 是有序图，我们证明了限制于 [math] 无序图的有序图列表-3-着色问题的近乎完整的二分法。特别是，我们证明了如果 [math] 最多只有一条边，那么这个问题可以在多项式时间内求解；如果 [math] 至少有三条边，那么这个问题仍然是 NP-完全的。此外，在 [math] 恰好有两条边的情况下，当 [math] 的两条边共用一个末端时，我们给出了一个完整的二分法，而当 [math] 的两条边不共用一个末端时，我们证明了几个 NP-完备性结果，从而将开放情况缩小到三种非常特殊的双边有序图。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.