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### First Fit Coloring of Interval Graphs

William T. Trotter

Georgia Institute of Technology

October 14, 2005

Interval Graphs are Perfect

Χ = ω = 4

On-Line Coloring of Interval Graphs

Suppose the vertices of an interval graph are presented one at a time by a Graph Constructor. In turn, Graph Colorer must assign a legitimate color to the new vertex. Moves made by either player are irrevocable.

Optimal On-Line Coloring

- Theorem (Kierstead and Trotter, 1982)
- There is an on-line algorithm that will use at most 3k-2 colors on an interval graph G for which the maximum clique size is at most k.
- This result is best possible.
- The algorithm does not need to know the value of k in advance.
- The algorithm is not First Fit.
- First Fit does worse when k is large.

How Well Does First Fit Do?

- For each positive integer k, let FF(k) denote the largest integer t for which First Fit can be forced to use t colors on an interval graph G for which the maximum clique size is at most k.
- Woodall (1976) FF(k) = O(k log k).

Academic Algorithm - Rules

- A Belongs to an interval
- B Left neighbor is A
- C Right neighbor is A
- D Some terminal set of letters

has more than 25% A’s

- F All else fails.

The Piercing Lemma

Lemma: Every interval J is pierced by a column of passing grades.

Proof: We use a double induction. Suppose the interval J is at level j. We show that for every i = 1, 2, …, j, there is a column of grades passing at level i which is under interval J

Initial Segment Lemma

Lemma: In any initial segment of n letters all of which are passing,

a ≥ (n – b – c)/4

A Column Surviving at the End

- b ≤ n/4
- c ≤ n/4
- n ≥ h+3
- h ≤ 8a - 3

Lower Bounds on FF(k)

Theorem: Kierstead and Trotter (1982)

There exists ε > 0 so that

FF(k) ≥ (3 + ε)k

when k is sufficiently large.

Lower Bounds on FF(k)

Theorem: Chrobak and Slusarek (1990)

FF(k) ≥ 4.4 k

when k is sufficiently large.

Lower Bounds on FF(k)

Theorem: Kierstead and Trotter (2004)

FF(k) ≥ 4.99 k

when k is sufficiently large.

A Likely Theorem

Our proof that FF(k) ≥ 4.99 k is computer assisted. However, there is good reason to believe that we can actually write out a proof to show:

For every ε > 0, FF(k) ≥ (5 – ε) k when k is sufficiently large.

A Negative Result and a Conjecture

However, we have been able to show that the Tree-Like walls used by all authors to date in proving lower bounds will not give a performance ratio larger than 5. As a result it is natural to conjecture that

As k tends to infinity, the ratio FF(k)/k tends to 5.

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