[Paper Review] On color-critical ($P_{5},\overline{P}_5$)-free graphs

Harjinder S. Dhaliwal, Angèle M. Hamel+4 more|arXiv (Cornell University)|Mar 31, 2014

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper proves that for every fixed k, the number of k-critical (P₅, P̄₅)-free graphs is finite, establishing the existence of a certifying algorithm for k-coloring such graphs. The proof relies on structural decomposition using buoys, modules, and clique cutsets, showing that all k-critical (P₅, P̄₅)-free graphs arise from specific join constructions of smaller critical graphs or buoys with constrained patterns.

ABSTRACT

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical $P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical $(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result implies the existence of a certifying algorithm for $k$-coloring $(P_5, \overline{P}_5)$-free graphs.

Motivation & Objective

• To establish the existence of a certifying algorithm for k-coloring (P₅, P̄₅)-free graphs.
• To determine whether the number of k-critical (P₅, P̄₅)-free graphs is finite for each fixed k.
• To characterize the structure of k-critical (P₅, P̄₅)-free graphs using buoy decomposition and join operations.
• To compute exact counts of k-critical (P₅, P̄₅)-free graphs for k ≤ 8.

Proposed method

• Structural analysis of k-critical graphs using module decomposition and the absence of comparable vertices.
• Application of the fact that k-critical graphs cannot have a clique cutset unless they are complete graphs.
• Decomposition of k-critical (P₅, P̄₅)-free graphs into buoys formed by substituting critical graphs into a C₅ framework.
• Enumeration of valid patterns (k₁,…,k₅) satisfying kᵢ > 0, kᵢ + kᵢ₊₁ ≤ k−1, and ∑kᵢ = 2k−1 to generate all possible buoy structures.
• Construction of all k-critical (P₅, P̄₅)-free graphs as joins of smaller critical graphs or buoys, using the set operation ⊗.
• Computation of exact cardinalities via recursive counting over valid patterns and known critical graph sets Cₖ.

Experimental results

Research questions

• RQ1 Is the number of k-critical (P₅, P̄₅)-free graphs finite for every fixed k?
• RQ2 Can k-coloring of (P₅, P̄₅)-free graphs be decided by a certifying algorithm?
• RQ3 What structural properties characterize k-critical (P₅, P̄₅)-free graphs?
• RQ4 How many non-isomorphic k-critical (P₅, P̄₅)-free graphs exist for small values of k?
• RQ5 What is the role of buoy structures and pattern constraints in generating all k-critical (P₅, P̄₅)-free graphs?

Key findings

• The number of k-critical (P₅, P̄₅)-free graphs is finite for every fixed k, resolving an open question about certifying algorithms.
• For k=4, there are exactly 3 non-isomorphic 4-critical (P₅, P̄₅)-free graphs, including two from the join of K₁ and C₃, and one buoy structure.
• For k=5, there are exactly 9 non-isomorphic 5-critical (P₅, P̄₅)-free graphs, comprising 6 from buoy constructions and 3 from the join of K₁ and C₄.
• For k=6, there are exactly 31 non-isomorphic 6-critical (P₅, P̄₅)-free graphs, with 21 from buoy structures and 10 from join operations.
• The counts grow rapidly: |C₇| = 185 and |C₈| = 1,487, indicating exponential growth in the number of critical graphs with k.
• All k-critical (P₅, P̄₅)-free graphs are generated by joining smaller critical graphs or constructing buoys with valid pattern constraints.