-list Vertex Coloring in Linear Time San Skulrattanakulchai - - PowerPoint PPT Presentation

∆-list Vertex Coloring in Linear Time

San Skulrattanakulchai

skulratt@cs.colorado.edu

University of Colorado at Boulder Colorado, USA

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Topics

Introduction List Coloring Problems, Definitions

∆-list Coloring Problem

Previous Work, Contribution The Theorem Our Proof Algorithm Subcubic Graphs Conclusion Open Problems

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Introduction

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List Coloring Problem

Generalizes the usual vertex coloring problem Applications: channel assignment, traffic phasing Each vertex v has a list L(v) of admissible colors Each vertex chooses an admissible color; adjacent vertices must choose distinct colors At least as hard as the usual problem; likely harder (Erd˝

• s et al, 1979)

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Definitions

∆ ≡ maximum degree k-colorable ≡ can color using ≤ k colors

list-colorable ≡ can color using admissible colors from given lists L(·)

f-choosable ≡ can list-color if |L(v)| ≥ f(v) for all v d-choosable ≡ use vertex degrees d(·) as f k-choosable ≡ use f(v) = k for all v

to k-list color ≡ to list-color when given lists L(·) with

|L(v)| ≥ k for all v

Brooks graph ≡ connected, not complete, not odd cycle

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∆-list Coloring Problem

When possible, to list-color given graph every vertex v

• f which has color list L(v) of size ≥ ∆

Algorithm naturally specializes to ∆-Coloring Problem ≡ to color given graph using ≤ ∆ colors, when possible To specialize, simply set L(v) := {1, 2, . . . , ∆} for every v

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Previous Work

Brooks characterizes ∆-colorable graphs Lovász’s proof of Brooks’ Theorem gives ∆-coloring algorithm, using routines for computing 3-connectivity, biconnected components, and recoloring Erd˝

• s, Rubin & Taylor characterize ∆-choosable graphs

ERT’s proof checks biconnectivity after vertex/edge deletions in each recursive step, so algorithm is super-linear

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Contribution

Simpler, algorithmic proof of ERT’s Theorem First O(n + m) time & space ∆-list coloring algorithm Algorithm specializes to new ∆-coloring algorithm, simpler than Lovász’s but same resource bound Algorithm simplifies for 3-list coloring subcubic graphs (∆ = 3)

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The Theorem

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Erd˝

• s, Rubin & Taylor’s Theorem

Erd˝

• s et al. (1979):

A Brooks graph is ∆-choosable

Brooks (1941): A Brooks graph is ∆-colorable (Brooks graph ≡ connected, not complete, not odd cycle)

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Our Proof

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Proof Ideas

identify classes of graphs that are “easy-to-recognize” and “easy-to-color” (d-choosable) search the input graph for an occurence of any such induced subgraph use the well-known technique of “vertex ordering through spanning tree with backwards coloring” to complete the coloring

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Trivial Graph Coloring Lemma

{∗}

The trivial graph is 1-choosable

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Non-regular Graph Lemma

Lemma Connected & not regular ⇒ ∆-choosable

v1 v2 v3 v6 v5 v4

Proof Let v1 have d(v1) < ∆. Grow a spanning tree rooted at

• v1. Color the vertices bottom up.

(Our graph is actually d′-choosable, where d′ is the same as d except that d′(v1) = d(v1)+1)

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Important Graphs I

An even cycle has an even number of edges

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Important Graphs II

A whel is a wheel with ≥ 1 spoke missing and ≥ 2 spokes remaining

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Important Graphs III

A θ(a, b, c) graph has 3 internally-disjoint paths of lengths a, b, c connecting two end vertices, where a ≤ b ≤ c

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A Handy Important Graph

A diamond is the smallest whel and also the θ(1, 2, 2) graph

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Cycle List Coloring Lemma

{∗, ∗, . . .} {∗, ∗, . . .} {∗, ∗, . . .} {∗, ∗, . . .} {∗, ∗, . . .}

If each vertex of a cycle has ≥ 2 colors in its list then the cycle is list-colorable unless

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Cycle List Coloring Lemma (Cont.)

{a, b} {a, b} {a, b} {a, b} {a, b}

ODD the cycle is odd and all lists are the same 2-list

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Even Cycle Choosability Lemma

In particular,

{∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗}

EVEN an even cycle is d-choosable

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Whel Choosability Lemma

{∗, ∗, ∗} {∗, ∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗, ∗}

A whel is (better than) d-choosable

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θ Graph Choosability Lemma

{∗, ∗, ∗} {∗, ∗} {∗, ∗} {∗, ∗, ∗} {∗, ∗}

A θ graph is d-choosable

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d-choosable Subgraph Lemma

Lemma Connected & contains a d-choosable induced subgraph ⇒ d-choosable

v2 v3 v4 v5

Proof Let H be d-choosable induced subgraph. (Here the left diamond.) Contract H. Grow a spanning tree rooted at the contracted H. Color the vertices bottom up, except H. Expand H. Color H.

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Biconnected Brooks Lemma

Lemma Biconnected Brooks graph contains, as an induced subgraph, one of (1) even cycle (2) θ graph (3) whel Proof By biconnectivity, take an induced cycle C. There are 3 possibilities for C. (1) C is even. Done.

(Continue. . . )

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Biconnected Brooks Lemma (Cont.)

(2) C is a triangle. Look at a maximal clique K containing C. If some neighbor w of K is adjacent to > 1 vertex of K (left picture), then have a diamond, done.

w K K

Otherwise (right picture) take a shortest path avoiding K from one vertex of K to another vertex of K. Now have a θ graph.

(Continue. . . )

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Biconnected Brooks Lemma (Cont.)

(3) C is odd but not a triangle. If some neighbor w of C is adjacent to > 1 vertex of C (left picture), then have a whel, done.

w C C

Otherwise (right picture) take a shortest path avoiding C from one vertex of C to another vertex of C. Now have a

θ graph.

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Our Proof of ERT’s Theorem

Theorem (ERT) A Brooks graph is ∆-choosable Proof If non-regular, done. If regular, find an endblock H. (Any hatched block in the picture will do.) This block H is necessarily Brooks, so has a d-choosable induced sub-

• graph. Again done.

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Algorithm

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Algorithm

Key Idea To achieve linear time & space, chop all color lists to the sizes of (maybe 1 more than) vertex degrees

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Subcubic Graphs

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Decomposition Theorem

A subcubic graph can be decomposed into edge-disjoint subgraphs C and F, where C is a collection of vertex-disjoint cycles and F is a forest of maximum degree no bigger than 3. Furthermore, cycles in C can be chosen to be

• induced. (See Skulrattanakulchai IPL (2002) and Gabow & Skulrattanakulchai

COCOON’02.)

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3-list-coloring

If G is not cubic, then done. Else G is cubic. Decompose G into induced cycles C and forest F. If C has some even cycle then done. Else if there is a vertex w adjacent to > 1 vertex of some C ∈ C (left picture)

C w

C C’

then have a whel, done. Else if there are cycles C, C′ ∈ C joined by 2 edges (right picture) then have a θ graph, done. Else contract all cycles in C and find an induced cycle D in the contracted graph (next page picture).

(Continue. . . )

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3-list-coloring (Cont.)

D

Replace (intelligently) each vertex in D corresponding to a contracted cycle by one of the two paths (in the original graph) of the contracted cycle. Now have an induced even cycle or an induced θ graph, done.

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Conclusion

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Conclusion

Let G = (V, E) be an undirected graph without self-loops, every vertex of which has been assigned its own list (set) of admissible colors. Suppose that all the following conditions are satisfied.

• 1. G contains no (∆ + 1)-clique, and contains no odd cycle

if ∆ = 2.

• 2. Every vertex v has ≥ d(v) colors in its list, with strict

inequality holding for at least one vertex in each connected component that is not regular. Then it is possible, in O(|V |+|E|) time and space, to choose for each vertex a color from its list so that adjacent vertices receive distinct colors.

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Open Problems

∆-list-coloring in NC? d-choosability in NC?

Algorithms for extentions of ERT’s theorem?

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