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The (Parameterized) Complexity of Equitable Coloring The (Parameterized) complexity of Equitable Coloring Guilherme C. M. Gomes Matheus R. Guedes Vinicius F. dos Santos Carlos Vinicius G. C. Lima Universidade Federal de Minas Gerais Grafos e Combinatória - 1a temporada Equitable Coloring 1 / 34 Equitable Coloring Equitable Coloring Grafos e Combinatória - 1a temporada Equitable Coloring 2 / 34 Equitable Coloring Equitable (Vertex) Coloring Can we k-color G such that the size of two color classes differ by ≤ 1? Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring Can we k-color G such that the size of two color classes differ by ≤ 1? Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring Can we k-color G such that the size of two color classes differ by ≤ 1? Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring Can we k-color G such that the size of two color classes differ by ≤ 1? Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Applications • Municipal garbage collection; • Processor task scheduling; • Communication control; • Server load balancing. Grafos e Combinatória - 1a temporada Equitable Coloring 4 / 34 Equitable Coloring Conjecture For every connected graph G which is neither a complete graph nor an odd-hole, χ=(G) ≤ ∆(G). Equitable Coloring Some important stuff I Equitable chromatic number The smallest integer k such that G is equitably k-colorable is the equitable chromatic number χ=(G). Grafos e Combinatória - 1a temporada Equitable Coloring 5 / 34 Equitable Coloring Some important stuff I Equitable chromatic number The smallest integer k such that G is equitably k-colorable is the equitable chromatic number χ=(G). Equitable Coloring Conjecture For every connected graph G which is neither a complete graph nor an odd-hole, χ=(G) ≤ ∆(G). Grafos e Combinatória - 1a temporada Equitable Coloring 5 / 34 Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1. Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds. Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248 Equitable Coloring Some important stuff II Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G). Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds. Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248 Equitable Coloring Some important stuff II Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G). Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1. Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248 Equitable Coloring Some important stuff II Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G). Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1. Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds. Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable Coloring Some important stuff II Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G). Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1. Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds. Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248 Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable Coloring The story so far... Class Complexity Trees P Forests P Bipartite NP-complete, even if k = 3 Co-bipartite P Cographs NP-complete, P for each fixed k Bounded Treewidth P Chordal NP-complete Block ? Split P Unipolar ? Interval NP-complete Co-interval P Grafos e Combinatória - 1a temporada Equitable Coloring 7 / 34 Equitable Coloring After this talk Class Complexity Trees P Forests P Bipartite NP-complete, even if k = 3 Co-bipartite P Cographs NP-complete, P for each fixed k Bounded Treewidth P Chordal NP-complete Block NP-complete Split P Unipolar P Interval NP-complete Co-interval P Grafos e Combinatória - 1a temporada Equitable Coloring 8 / 34 a1 For each item of A, build a gad- get with some key vertices. a2 All key vertices must have the same color. a3 Key vertices with color i → item a4 in i-th bin. 0 1 2 3 4 Equitable Coloring Bin Packing P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B? Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 For each item of A, build a gad- get with some key vertices. All key vertices must have the same color. Key vertices with color i → item in i-th bin. Equitable Coloring Bin Packing P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B? a1 a2 a3 a4 0 1 2 3 4 Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 Equitable Coloring Bin Packing P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B? a1 For each item of A, build a gad- get with some key vertices. a2 All key vertices must have the same color. a3 Key vertices with color i → item a4 in i-th bin. 0 1 2 3 4 Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 a1 a2 y 1 y2 a3 a4 0 1 2 3 4 y y4 3 k = 4 Equitable Coloring (a, k)-flowers Create a + 1 cliques with k − 1 vertices and add one universal vertex. Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 y 1 y2 y y4 3 Equitable Coloring (a, k)-flowers Create a + 1 cliques with k − 1 vertices and add one universal vertex. a1 a2 a3 a4 0 1 2 3 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 Equitable Coloring (a, k)-flowers Create a + 1 cliques with k − 1 vertices and add one universal vertex. a1 a2 y 1 y2 a3 a4 0 1 2 3 4 y y4 3 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins bin1 bin2 bin3 y bin4 0 1 2 3 4 aj = 2, k = 4 k = 4 Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n] X = (k − 1) n + aj + n j∈[n] = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n) Equitable Coloring Block Graphs Theorem equitable coloring of block graphs is NP-complete. Construct a graph G as the disjoint union of flowers Fj = F (aj , k) and try to equitably k-color it. Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 X = (k − 1) n + aj + n j∈[n] = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n) Equitable Coloring Block Graphs Theorem equitable coloring of block graphs is NP-complete. Construct a graph G as the disjoint union of flowers Fj = F (aj , k) and try to equitably k-color it. X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n] Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n) Equitable Coloring Block Graphs Theorem equitable coloring of block graphs is NP-complete. Construct a graph G as the disjoint union of flowers Fj = F (aj , k) and try to equitably k-color it.
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