# A Note on k-Colorability of P5-Free Graphs

TL;DR: A polynomial-time algorithm determining whether or not, for a fixed k, a P 5 -free graph can be k-colored is presented, and if such a coloring exists, the algorithm will produce one.

Abstract: We present a polynomial-time algorithm determining whether or not, for a fixed k, a P 5 -free graph can be k-colored. If such a coloring exists, the algorithm will produce one.

## Summary (1 min read)

### 1 Introduction

- Graph coloring is among the most important and applicable graph problems.
- For other classes of graphs, like perfect graphs [8], the problem is polynomial-time solvable.
- Then in Section 3, the authors present their recursive polynomial-time algorithm that answers thek-colorability question forP5-free graphs.

### 2 Background and Definitions

- In this section the authors provide the necessary background and definitions used in the rest of the paper.
- The following structural result aboutP5-free graphs is from Bacsó and Tuza [2]: THEOREM 1 Every connected P5-free graph has either a dominating clique or a dominating P3.
- This listing corresponds to their initial instanceΦ.

### 3 The Algorithm

- This section describes a polynomial time algorithm that decides whether or notG is k-colorable.
- Identify and color a maximal dominating clique or aP3 if no such clique exists (Theorem 1).
- This partitions the vertices intofixed setsindexed by available colors.
- Thus, for each instanceφi the authors recursively see if each fixed set can be colored with the corresponding restricted color lists (the base case is when the color listsare a single color).
- As mentioned, the difficult part is reducing the dependencies between each pair of fixed sets (Step 2).

### 3.1 Removing the Dependencies Between Two Fixed Sets

- Let Slist denote a fixed set of vertices with color list given bylist.
- This is becauseP and Q are subsets of different fixed sets.
- Thus there must be at most one special componentC.
- Using this procedure along with Theorem 2, the authors can remove the dep ndencies between two dynamic setsP andQ for a given list-coloring instanceφ.
- Since the authors know that the special componentC has already been handled.

### 4 Summary

- The algorithm recursively uses list coloring techniques and thus its complexity is high even though it is polynomial, as is the case with all list coloring algorithms.
- In a relatedpaper (in preparation), the authors will give a slightly faster algorithm also based on list coloring techniques, however this algorithm provides less insight into the structure ofP5-free graphs.
- Is the problem ofk-coloring aP7-free graph NP-complete for anyk ≥.
- Two other related open problems are to determine the complexities of theMAXIMUM INDEPENDENT SET andMINIMUM INDEPENDENT DOMINATING SET problems onP5-free graphs.

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##### Citations

44 citations

### Cites background or methods from "A Note on k-Colorability of P5-Free..."

...Previously known algorithms ([5,10,13]) provide a yes-certificate by constructing a 3-coloring if one exists....

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...In [4] and [5], it is shown that k-COLORABILITY can be solved for the class of P5-free graphs in polynomial time for every particular value of k....

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...Besides [4], there are several polynomial-time algorithms for 3-coloring a P5-free graph ([5,10,13]) but none of them is a certifying algorithm....

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34 citations

### Cites background from "A Note on k-Colorability of P5-Free..."

...Finding the chromatic number of a P5-free graphs is NP-hard [13], but for every fixed k, the problemof coloring a graphwith k colors admits a polynomial-time algorithm [11,12]....

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27 citations

17 citations

13 citations

### Cites background from "A Note on k-Colorability of P5-Free..."

...An interesting point to mention is that the fixed parameter tractability of k-Coloring on P5free graphs is still open [ 10 ]....

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...In last year’s MFCS, Ho`ang et al. showed that k-Coloring can be solved in polynomial time for any fixed k on P5-free graphs [ 10 ], but in their running time k contributes to the degree of the polynomial....

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##### References

8,634 citations

6,663 citations

### Additional excerpts

...Ingeneral, thek-colorability problem is NPcomplete [10]....

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...In general, thek-colorability problem is NPcomplete [10]....

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4,086 citations

### "A Note on k-Colorability of P5-Free..." refers background in this paper

...For more information on perfect graphs, see [1], [3], and [7]....

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2,062 citations

2,060 citations

### Additional excerpts

...Keywords: P5-free, graph coloring, dominating clique...

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