Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems On the Complexity of Defective Coloring Rémy Belmonte 1 Michael Lampis 2 Valia Mitsou 3 1 University of Electro-Communications, Tokyo 2 Lamsade, CNRS and Université Paris-Dauphine, Paris 3 Liris, CNRS and Université Lyon 1, Lyon June 20, 2017 1/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Definitions and motivation 2/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems (Defective) Coloring Definition (Proper coloring) A proper (vertex) c -coloring of a graph G is an assignment of colors { 1 , . . . , c } to the vertices of G such that no two adjacent vertices are assigned the same color. Figure: Petersen graph 3/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems (Defective) Coloring Definition (Proper coloring) A proper (vertex) c -coloring of a graph G is an assignment of colors { 1 , . . . , c } to the vertices of G such that no two adjacent vertices are assigned the same color. Figure: A proper 3-coloring 3/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems (Defective) Coloring Definition (Defective coloring) A defective ( c , ∆ ∗ ) -coloring of a graph G is an assignment of colors { 1 , . . . , c } to the vertices of G such that every vertex of G has at most ∆ ∗ neighbors sharing its color. Figure: A defective (2,1)-coloring 3/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Applications: Radio networks Antennas which are within each-other’s range must be assigned different frequencies. 4/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Applications: Radio networks Antennas which are within each-other’s range must be assigned different frequencies. 4/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Applications: Radio networks Antennas which are within each-other’s range must be assigned different frequencies. If we can afford signal interference from at most ∆ ∗ = 1 neighboring antenna, we can use less frequencies. 4/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - simple facts For any graph G with n vertices: A proper c -coloring of G is a defective ( c , 0 ) -coloring of G . 5/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - simple facts For any graph G with n vertices: A proper c -coloring of G is a defective ( c , 0 ) -coloring of G . Defective coloring is already NP-hard for c = 3 and ∆ ∗ = 0. 5/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - simple facts For any graph G with n vertices: A proper c -coloring of G is a defective ( c , 0 ) -coloring of G . Defective coloring is already NP-hard for c = 3 and ∆ ∗ = 0. There always exists an ( 1 , ∆) -coloring of G , where ∆ is its maximum degree. 5/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - simple facts For any graph G with n vertices: A proper c -coloring of G is a defective ( c , 0 ) -coloring of G . Defective coloring is already NP-hard for c = 3 and ∆ ∗ = 0. There always exists an ( 1 , ∆) -coloring of G , where ∆ is its maximum degree. If c · k ≥ n then G has a ( c , k − 1 ) -coloring. Figure: Any graph on n vertices is ( c , ⌈ n / c ⌉ − 1 ) -colorable 5/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]: 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]: For any ∆ ∗ ≥ 1, ( 2 , ∆ ∗ ) -coloring is NP-complete on planar graphs. 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]: For any ∆ ∗ ≥ 1, ( 2 , ∆ ∗ ) -coloring is NP-complete on planar graphs. ( 3 , 1 ) -coloring is NP-complete on planar graphs. 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]: For any ∆ ∗ ≥ 1, ( 2 , ∆ ∗ ) -coloring is NP-complete on planar graphs. ( 3 , 1 ) -coloring is NP-complete on planar graphs. [Havet, Kang, and Sereni 2009]: For any c ≥ 2 , ∆ ∗ ≥ 1, ( c , ∆ ∗ ) -coloring is NP-complete on unit-disk graphs. 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Defective Coloring - previous work Independently introduced by [Andrews and Jacobson 1985] and [Cowen, Cowen, and Woodall 1986]. [Cowen, Goddard, Jesurum 1997]: For any ∆ ∗ ≥ 1, ( 2 , ∆ ∗ ) -coloring is NP-complete on planar graphs. ( 3 , 1 ) -coloring is NP-complete on planar graphs. [Havet, Kang, and Sereni 2009]: For any c ≥ 2 , ∆ ∗ ≥ 1, ( c , ∆ ∗ ) -coloring is NP-complete on unit-disk graphs. [Angelini et al. 2017]: ( 3 , 1 ) -coloring is NP-complete on graphs of degree at most 6 and on planar graphs of degree at most 7. 6/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Coloring Perfect Graphs Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number. 7/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Coloring Perfect Graphs Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number. Theorem (Proper) coloring is polynomially solvable on perfect graphs. 7/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Coloring Perfect Graphs Definition A graph G is called perfect when for every induced subgraph, its chromatic number equals its clique number. Theorem (Proper) coloring is polynomially solvable on perfect graphs. Motivation Study defective coloring on classes of perfect graphs. 7/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Coloring Perfect Graphs Chordal graphs Cographs NP-hard on Split if c ≥ 2 NP-hard NP-hard on Split if ∆ ∗ ≥ 1 In P if c or ∆ ∗ is fixed � n 4 / 5 � In P if c , ∆ ∗ fixed O Solvable in n In P on Trivially perfect for any c , ∆ ∗ 7/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Chordal Graphs 8/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Chordal Graphs - hardness Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. 9/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Chordal Graphs - hardness Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. 9/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Chordal Graphs - hardness Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. Fact Split graphs are chordal. 9/27
Definitions and motivation Chordal Graphs Cographs Trivially perfect graphs Summary - Open problems Chordal Graphs - hardness Definition A graph G is chordal if all cycles of length ≥ 4 have a chord. Definition A graph G is split if its vertices can be partitioned into a clique and an independent set. Fact Split graphs are chordal. Theorem Defective coloring is NP-hard on split graphs for any ∆ ∗ ≥ 1 . 9/27
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