Coloring Algorithms on Subcubic Graphs Harold N. Gabow, San - PowerPoint PPT Presentation

Coloring Algorithms on Subcubic Graphs Harold N. Gabow, San Skulrattanakulchai hal@cs.colorado.edu, skulratt@cs.colorado.edu University of Colorado at Boulder Colorado, USA Coloring Algorithms on Subcubic Graphs – p.1/56

Introduction Graph Coloring To color a graph ≡ to assign color to vertices/edges so that no adjacent/incident vertices/edges receive the same color Flavors: vertex, edge, total, list coloring Why subcubic graphs (∆ = 3) ? some problems too difficult on general graphs some problems have linear-time reduction to subcubic graphs some “real-world” applications are on subcubic graphs Coloring Algorithms on Subcubic Graphs – p.2/56

Notation & NP-Hardness Notation (list) chromatic number χ ( χ ℓ ) (list) edge chromatic number χ ′ ( χ ′ ℓ ) (list) total chromatic number χ ′′ ( χ ′′ ℓ ) NP -hardness Vertex Coloring—Karp (1972) from 3-SAT Edge Coloring—Holyer (1981) from 3-SAT Total Coloring—Sánchez-Arroyo (1989) from Edge Coloring Coloring Algorithms on Subcubic Graphs – p.3/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? List Total Coloring Conjecture χ ′′ ℓ = χ ′′ ? Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? List Total Coloring Conjecture χ ′′ ℓ = χ ′′ ? χ ℓ � = χ in general! 1,2 • • 1,2 ❜ ✧ ❡ ✪ ❜ ✧ ✧ ❜ ✧ ❜ ❡ ✪ 2,3 • • 2,3 ❜ ✧ ✪ ❡ ❜ ✧ ✧ ❜ ✪ ✧ ❜ ❡ 3,1 • • 3,1 Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? List Total Coloring Conjecture χ ′′ ℓ = χ ′′ ? χ ℓ � = χ in general! χ ′ ℓ ≤ ∆ + 1 if ∆ = 3 , 4. LCC holds if graph is bipartite, or series-parallel, or line-perfect, or a multicircuit Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? List Total Coloring Conjecture χ ′′ ℓ = χ ′′ ? χ ℓ � = χ in general! χ ′ ℓ ≤ ∆ + 1 if ∆ = 3 , 4. LCC holds if graph is bipartite, or series-parallel, or line-perfect, or a multicircuit TCC holds if graph is bipartite, or complete r -partite, or ∆ = 3 , 4, 5, or ∆ ≥ n − 5 , or ∆ ≥ (3 / 4) n Coloring Algorithms on Subcubic Graphs – p.4/56

Conjectures & Some Known Facts χ ′ ℓ = χ ′ ? List Coloring Conjecture (LCC) χ ′′ ≤ ∆ + 2 for simple Total Coloring Conjecture (TCC) graphs? List Total Coloring Conjecture χ ′′ ℓ = χ ′′ ? χ ℓ � = χ in general! χ ′ ℓ ≤ ∆ + 1 if ∆ = 3 , 4. LCC holds if graph is bipartite, or series-parallel, or line-perfect, or a multicircuit TCC holds if graph is bipartite, or complete r -partite, or ∆ = 3 , 4, 5, or ∆ ≥ n − 5 , or ∆ ≥ (3 / 4) n χ ′′ ℓ ≤ 5 if ∆ = 3 Coloring Algorithms on Subcubic Graphs – p.4/56

Contributions Decomposition principle for subcubic graphs New, simpler proofs & linear-time algorithms for 4-edge-coloring (Skulrattanakulchai, IPL 81 (4) 2002, 191–195) 4-list-edge-coloring 5-total-coloring 5-list-total-coloring subcubic graphs. Algorithms shows subcubic graphs satisfy χ ′ ≤ 4 , χ ′ ℓ ≤ 4 , χ ′′ ≤ 5 , χ ′′ ℓ ≤ 5 . The first two are the simplest known, the last two are the first linear-time algorithms. O ( n/ log n ) processors, O (log n ) time whp EREW PRAM algorithm for 4-list-edge-coloring subcubic graphs Coloring Algorithms on Subcubic Graphs – p.5/56

Greedy Coloring? An edge can have up to 4 neighboring edges and 2 neighboring vertices. A vertex can have up to 3 neighboring edges and 3 neighboring vertices. Coloring Algorithms on Subcubic Graphs – p.6/56

Greedy Coloring? An edge can have up to 4 neighboring edges and 2 neighboring vertices. A vertex can have up to 3 neighboring edges and 3 neighboring vertices. So simple-minded greedy coloring fails. Coloring Algorithms on Subcubic Graphs – p.6/56

Decomposition Theorem Coloring Algorithms on Subcubic Graphs – p.7/56

Coloring Algorithms on Subcubic Graphs – p.8/56

Decomposition Theorem A subcubic graph G can be decomposed into edge-disjoint subgraphs C and T , where C is a collection of vertex-disjoint cy- cles and T is a forest of maximum degree no bigger than 3. Furthermore, G admits a decomposition without chords unless it contains a triple bond. Coloring Algorithms on Subcubic Graphs – p.9/56

Tree Edge Coloring Lemma Coloring Algorithms on Subcubic Graphs – p.10/56

3-edge-coloring Subcubic Tree Coloring Algorithms on Subcubic Graphs – p.11/56

3-edge-coloring Subcubic Tree 1 Coloring Algorithms on Subcubic Graphs – p.12/56

3-edge-coloring Subcubic Tree 2 1 Coloring Algorithms on Subcubic Graphs – p.13/56

3-edge-coloring Subcubic Tree 2 1 2 Coloring Algorithms on Subcubic Graphs – p.14/56

3-edge-coloring Subcubic Tree 2 1 2 3 Coloring Algorithms on Subcubic Graphs – p.15/56

3-edge-coloring Subcubic Tree 2 3 1 2 3 Coloring Algorithms on Subcubic Graphs – p.16/56

3-edge-coloring Subcubic Tree 2 3 1 1 2 3 Coloring Algorithms on Subcubic Graphs – p.17/56

Tree Edge Coloring Lemma Conclusion: A subcubic tree is 3-list-edge-colorable. Coloring Algorithms on Subcubic Graphs – p.18/56

Tree Total Coloring Lemma Coloring Algorithms on Subcubic Graphs – p.19/56

4-total-coloring Subcubic Tree Coloring Algorithms on Subcubic Graphs – p.20/56

4-total-coloring Subcubic Tree 1 Coloring Algorithms on Subcubic Graphs – p.21/56

4-total-coloring Subcubic Tree 1 2 Coloring Algorithms on Subcubic Graphs – p.22/56

4-total-coloring Subcubic Tree 1 2 3 Coloring Algorithms on Subcubic Graphs – p.23/56

4-total-coloring Subcubic Tree 1 3 2 3 Coloring Algorithms on Subcubic Graphs – p.24/56

4-total-coloring Subcubic Tree 1 3 2 3 2 Coloring Algorithms on Subcubic Graphs – p.25/56

4-total-coloring Subcubic Tree 4 1 3 2 3 2 Coloring Algorithms on Subcubic Graphs – p.26/56

4-total-coloring Subcubic Tree 2 4 1 3 2 3 2 Coloring Algorithms on Subcubic Graphs – p.27/56

4-total-coloring Subcubic Tree 1 2 4 1 3 2 3 2 Coloring Algorithms on Subcubic Graphs – p.28/56

4-total-coloring Subcubic Tree 3 1 2 4 1 3 2 2 3 Coloring Algorithms on Subcubic Graphs – p.29/56

4-total-coloring Subcubic Tree 3 1 2 3 4 1 3 2 2 3 Coloring Algorithms on Subcubic Graphs – p.30/56

4-total-coloring Subcubic Tree 3 1 2 3 4 1 1 3 2 2 3 Coloring Algorithms on Subcubic Graphs – p.31/56

4-total-coloring Subcubic Tree 3 1 2 3 4 1 1 2 3 2 2 3 Coloring Algorithms on Subcubic Graphs – p.32/56

4-total-coloring Subcubic Tree 3 1 2 3 4 1 1 2 3 2 3 2 3 Coloring Algorithms on Subcubic Graphs – p.33/56

Tree Total Coloring Lemma Conclusion: A subcubic tree is 4-list-total-colorable. Coloring Algorithms on Subcubic Graphs – p.34/56

Cycle Coloring Lemma I Coloring Algorithms on Subcubic Graphs – p.35/56

Cycle Coloring Lemma I >= 2 {1, 2} >= 2 >= 2 {1,2} {1,2} ODD >= 2 >= 2 {1,2} {1,2} >= 2 COLORABLE unless Odd & Same 2−list Coloring Algorithms on Subcubic Graphs – p.36/56

Cycle Coloring Lemma I Vertex Version (CVCL I) A cycle C where every vertex has a list of ≥ 2 colors is vertex-colorable unless C is odd and every list is the same list of 2 colors. Edge Version (CECL I) . . . Coloring Algorithms on Subcubic Graphs – p.37/56

Cycle Coloring Lemma II Coloring Algorithms on Subcubic Graphs – p.38/56

Cycle Coloring Lemma II {1} {2,3} {1,2} ODD {1,2} {1,2} Colorable Coloring Algorithms on Subcubic Graphs – p.39/56

Cycle Coloring Lemma II Vertex Version (CVCL II) Suppose the first vertex v 1 of an odd cycle C has a list of 1 color, vertex v 2 , . . . , v k − 1 has the same list of 2 colors, and the last vertex v k has a list of 2 colors. Suppose also that the list of v 1 is included in the list of v 2 , and the list of v k is not the same as the list of v 2 . Then C is vertex-colorable by colors from these lists. Edge Version (CECL II) . . . Coloring Algorithms on Subcubic Graphs – p.40/56

Cycle Total Coloring Lemma Coloring Algorithms on Subcubic Graphs – p.41/56