[Paper Review] Three-coloring graphs with no induced seven-vertex path I : the triangle-free case

Maria Chudnovsky, Peter Maceli+1 more|arXiv (Cornell University)|Sep 17, 2014

Advanced Graph Theory Research9 references16 citations

TL;DR

This paper presents a polynomial-time algorithm to determine 3-colorability in triangle-free graphs with no induced seven-vertex path (P7-free). By leveraging structural decomposition, palette updating, and reduction to 2-SAT, the algorithm efficiently tests for 3-colorings via exhaustive case analysis on 5-cycles and controlled subpalette enumeration, achieving O(|V|¹⁸) runtime for the full class of P7- and C3-free graphs.

ABSTRACT

In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.

Motivation & Objective

• To solve the 3-coloring problem for triangle-free graphs with no induced P7.
• To extend known polynomial-time results for P5- and P6-free graphs to the P7-free case.
• To address the open problem of 3-coloring in Pℓ-free graphs for ℓ ≥ 7.
• To develop a structural algorithmic framework for H-free graphs where H is a path.
• To provide a constructive algorithm that either produces a 3-coloring or determines non-existence in polynomial time.

Proposed method

• Uses a clean graph transformation to simplify structural analysis.
• Applies a recursive decomposition based on 5-cycles (C5), 7-cycles (C7), and shells to partition the vertex set.
• Employs palette updating to reduce coloring choices based on fixed-color vertices.
• Reduces the problem to multiple instances of 2-SAT using subpalette enumeration.
• Applies a 2-SAT solver (via Aspvall-Plass-Tarjan) to test colorability of palettes with at most two colors per vertex.
• Combines exhaustive case analysis over 3-colorings of a 5-cycle with subpalette pruning to bound the search space.

Experimental results

Research questions

• RQ1 Can 3-colorability be decided in polynomial time for triangle-free P7-free graphs?
• RQ2 What structural properties of P7-free triangle-free graphs enable efficient coloring?
• RQ3 How can palette updating and subpalette enumeration be used to reduce the search space in 3-coloring?
• RQ4 Can the 3-coloring problem for P7-free graphs be reduced to polynomially many 2-SAT instances?
• RQ5 What role do 5-cycles, shells, and 7-cycles play in the algorithmic decomposition of such graphs?

Key findings

• The algorithm runs in O(|V|¹⁸) time for {P7, C3}-free graphs, establishing polynomial-time solvability.
• For {P7, C3, C7, shell}-free graphs containing a 5-cycle, the algorithm runs in O(|V|⁶) time.
• The algorithm uses a finite set of subpalettes (O(|V|⁴)) derived from 5-cycle colorings to test global 3-colorability.
• Palette updating ensures that all vertex palettes have size at most 2 after initial constraints are applied.
• The algorithm correctly determines 3-colorability by testing all 53 possible 3-colorings of a 5-cycle and their compatible extensions.
• The solution relies on the fact that P7-free, triangle-free graphs without C7 or shells are bipartite, allowing 2-coloring in O(|V|³) time.