∆ -list Vertex Coloring in Linear Time San Skulrattanakulchai skulratt@cs.colorado.edu University of Colorado at Boulder Colorado, USA ∆ -list Vertex Coloring in Linear Time – p.1/37
Topics Introduction List Coloring Problems, Definitions ∆ -list Coloring Problem Previous Work, Contribution The Theorem Our Proof Algorithm Subcubic Graphs Conclusion Open Problems ∆ -list Vertex Coloring in Linear Time – p.2/37
Introduction ∆ -list Vertex Coloring in Linear Time – p.3/37
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˝ os et al, 1979) ∆ -list Vertex Coloring in Linear Time – p.4/37
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 ∆ -list Vertex Coloring in Linear Time – p.5/37
∆ -list Coloring Problem When possible, to list-color given graph every vertex v of 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 ∆ -list Vertex Coloring in Linear Time – p.6/37
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˝ os, Rubin & Taylor characterize ∆ -choosable graphs ERT’s proof checks biconnectivity after vertex/edge deletions in each recursive step, so algorithm is super-linear ∆ -list Vertex Coloring in Linear Time – p.7/37
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 ) ∆ -list Vertex Coloring in Linear Time – p.8/37
The Theorem ∆ -list Vertex Coloring in Linear Time – p.9/37
Erd˝ os, Rubin & Taylor’s Theorem Erd˝ os et al. (1979): A Brooks graph is ∆ -choosable Brooks (1941): A Brooks graph is ∆ -colorable (Brooks graph ≡ connected, not complete, not odd cycle) ∆ -list Vertex Coloring in Linear Time – p.10/37
Our Proof ∆ -list Vertex Coloring in Linear Time – p.11/37
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 ∆ -list Vertex Coloring in Linear Time – p.12/37
Trivial Graph Coloring Lemma {∗} The trivial graph is 1 -choosable ∆ -list Vertex Coloring in Linear Time – p.13/37
Non-regular Graph Lemma Lemma Connected & not regular ⇒ ∆ -choosable v 4 v 5 v 1 v 6 v 2 v 3 Proof Let v 1 have d ( v 1 ) < ∆ . Grow a spanning tree rooted at v 1 . Color the vertices bottom up. (Our graph is actually d ′ -choosable, where d ′ is the same as d except that d ′ ( v 1 ) = d ( v 1 )+1) ∆ -list Vertex Coloring in Linear Time – p.14/37
Important Graphs I An even cycle has an even number of edges ∆ -list Vertex Coloring in Linear Time – p.15/37
Important Graphs II A whel is a wheel with ≥ 1 spoke missing and ≥ 2 spokes remaining ∆ -list Vertex Coloring in Linear Time – p.16/37
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 ∆ -list Vertex Coloring in Linear Time – p.17/37
A Handy Important Graph A diamond is the smallest whel and also the θ (1 , 2 , 2) graph ∆ -list Vertex Coloring in Linear Time – p.18/37
Cycle List Coloring Lemma {∗ , ∗ , . . . } {∗ , ∗ , . . . } {∗ , ∗ , . . . } {∗ , ∗ , . . . } {∗ , ∗ , . . . } If each vertex of a cycle has ≥ 2 colors in its list then the cycle is list-colorable unless ∆ -list Vertex Coloring in Linear Time – p.19/37
Cycle List Coloring Lemma (Cont.) { a, b } { a, b } ODD { a, b } { a, b } { a, b } the cycle is odd and all lists are the same 2-list ∆ -list Vertex Coloring in Linear Time – p.20/37
Even Cycle Choosability Lemma In particular, {∗ , ∗} {∗ , ∗} {∗ , ∗} {∗ , ∗} EVEN {∗ , ∗} {∗ , ∗} an even cycle is d -choosable ∆ -list Vertex Coloring in Linear Time – p.21/37
Whel Choosability Lemma {∗ , ∗ , ∗} {∗ , ∗ , ∗} {∗ , ∗} {∗ , ∗} {∗ , ∗ , ∗} {∗ , ∗} A whel is (better than) d -choosable ∆ -list Vertex Coloring in Linear Time – p.22/37
θ Graph Choosability Lemma {∗ , ∗} {∗ , ∗ , ∗} {∗ , ∗ , ∗} {∗ , ∗} {∗ , ∗} A θ graph is d -choosable ∆ -list Vertex Coloring in Linear Time – p.23/37
d -choosable Subgraph Lemma Lemma Connected & contains a d -choosable induced subgraph ⇒ d -choosable v 3 v 4 v 5 v 2 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 . ∆ -list Vertex Coloring in Linear Time – p.24/37
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. . . ) ∆ -list Vertex Coloring in Linear Time – p.25/37
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. . . ) ∆ -list Vertex Coloring in Linear Time – p.26/37
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. ∆ -list Vertex Coloring in Linear Time – p.27/37
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. ∆ -list Vertex Coloring in Linear Time – p.28/37
Algorithm ∆ -list Vertex Coloring in Linear Time – p.29/37
Algorithm Key Idea To achieve linear time & space, chop all color lists to the sizes of (maybe 1 more than) vertex degrees ∆ -list Vertex Coloring in Linear Time – p.30/37
Subcubic Graphs ∆ -list Vertex Coloring in Linear Time – p.31/37
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 .) ∆ -list Vertex Coloring in Linear Time – p.32/37
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) w C 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. . . ) ∆ -list Vertex Coloring in Linear Time – p.33/37
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. ∆ -list Vertex Coloring in Linear Time – p.34/37
Conclusion ∆ -list Vertex Coloring in Linear Time – p.35/37
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