2013/04/21 by Shenwei Huang, Huang, Shenwei · 4 citations
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.1304.5808
openalex publication_date 2013/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A graph is H-free if it does not contain an induced subgraph isomorphic to H. We denote by Pk and Ck the path and the cycle on k vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P7-free graphs, and that 5-COLORING is NP-complete for P6-free graphs. These two results improve all previous results on k-coloring Pt-free graphs, and almost complete the classification of complexity of k-COLORING Pt-free graphs for k≥ 4 and t≥ 1, leaving as the only missing case 4-COLORING P6-free graphs. We expect that 4-COLORING is polynomial time solvable for P6-free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P6-free graphs which are also P-free, where P is the graph obtained from C4 by adding a new vertex and making it adjacent to exactly one vertex on the C4.