Nordhaus-Gaddum inequalities for coloring games Cl ement - PowerPoint PPT Presentation

Nordhaus-Gaddum inequalities for coloring games Cl´ ement Charpentier Joint work with Simone Dantas (Universidad Federal Fluminense), Celina M. H. de Figueiredo (Universidad Federal do Rio de Janeiro), Ana Furtado (Universidad Federal do Rio de Janeiro), Sylvain Gravier (Universit´ e Grenoble Alpes) GAG Workshop (Lyon, Oct. 2017) C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 1 / 38

Definitions Definitions Proper coloring A coloring of a graph is the assignment of a color to each vertex of the graph. A coloring is proper if two adjacent vertices have different colors. The chromatic number of a graph G is denoted by χ ( G ). C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 2 / 38

Definitions Definitions Coloring game The coloring game was introduced by Brahms in 1981 and rediscovered in 1991 by Bodlaender. At start : a graph G uncolored and a set Φ of colors. Alice and Bob take turns coloring an uncolored vertex of G with a color of Φ. Alice wins when the graph is fully colored. Bob wins if he can prevent Alice’s victory. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 3 / 38

Definitions Definitions Coloring game The coloring game was introduced by Brahms in 1981 and rediscovered in 1991 by Bodlaender. At start : a graph G uncolored and a set Φ of colors. Alice and Bob take turns coloring an uncolored vertex of G with a color of Φ. Alice wins when the graph is fully colored. Bob wins if he can prevent Alice’s victory. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 3 / 38

Definitions Definitions Coloring game The coloring game was introduced by Brahms in 1981 and rediscovered in 1991 by Bodlaender. At start : a graph G uncolored and a set Φ of colors. Alice and Bob take turns coloring an uncolored vertex of G with a color of Φ. Alice wins when the graph is fully colored. Bob wins if he can prevent Alice’s victory. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 3 / 38

Definitions Definitions Coloring game The coloring game was introduced by Brahms in 1981 and rediscovered in 1991 by Bodlaender. At start : a graph G uncolored and a set Φ of colors. Alice and Bob take turns coloring an uncolored vertex of G with a color of Φ. Alice wins when the graph is fully colored. Bob wins if he can prevent Alice’s victory. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 3 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 4 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 5 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 6 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 7 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 8 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 9 / 38

Definitions Example Coloring game A little game... C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 10 / 38

Definitions Monotony by subset ? Question If Alice has a winning strategy for k colors on a graph G , does she have a winning strategy for k colors on any subgraph of G ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 11 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 12 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 13 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 14 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 15 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 16 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 17 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 18 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 19 / 38

Definitions Coloring game Another game ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 20 / 38

Definitions Monotony by number of colors ? An open problem Let G be a graph on which Alice has a winning strategy with k colors. Let k ′ > k . Does Alice have a winning strategy for G with k ′ colors ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 21 / 38

Definitions Monotony by number of colors ? An open problem Let G be a graph on which Alice has a winning strategy with k colors. Let k ′ > k . Does Alice have a winning strategy for G with k ′ colors ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 21 / 38

Definitions Monotony by number of colors ? An open problem Let G be a graph on which Alice has a winning strategy with k colors. Let k ′ > k . Does Alice have a winning strategy for G with k ′ colors ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 21 / 38

Definitions Monotony by number of colors ? An open problem Let G be a graph on which Alice has a winning strategy with k colors. Let k ′ > k . Does Alice have a winning strategy for G with k ′ colors ? C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 21 / 38

Definitions Game chromatic number Coloring game The game chromatic number of a graph G is the smaller number of colors for which Alice has a winning strategy for G . Trivial bounds For any graph G , χ ( G ) ≤ χ g ( G ) ≤ ∆( G ) + 1. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 22 / 38

Definitions Game chromatic number Coloring game The game chromatic number of a graph G is the smaller number of colors for which Alice has a winning strategy for G . Trivial bounds For any graph G , χ ( G ) ≤ χ g ( G ) ≤ ∆( G ) + 1. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 22 / 38

Definitions Norhaus-Gaddum inequalities Theorem [Nordhaus and Gaddum, 1956] For any graph G of order n , 2 √ n ≤ χ ( G ) + χ ( G ) ≤ n + 1. Survey: [Aouiche and Hansen, 2013]. C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 23 / 38

Definitions Norhaus-Gaddum inequalities Result (1) Theorem [Nordhaus and Gaddum, 1956] For any graph G of order n , 2 √ n ≤ χ ( G ) + χ ( G ) ≤ n + 1. These bounds are tight for an infinite number of values of n . Theorem For any graph G of order n , 2 √ n ≤ χ g ( G ) + χ g ( G ) ≤ � 3 n � . 2 These bounds are asymptotically tight. For infinite values of n , there are graphs G such that √ χ g ( G ) + χ g ( G ) = 2 2 n − 1 and � 4 n � χ g ( G ) + χ g ( G ) = − 1. 3 C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 24 / 38

Definitions Norhaus-Gaddum inequalities Result (1) Theorem [Nordhaus and Gaddum, 1956] For any graph G of order n , 2 √ n ≤ χ ( G ) + χ ( G ) ≤ n + 1. These bounds are tight for an infinite number of values of n . Theorem For any graph G of order n , 2 √ n ≤ χ g ( G ) + χ g ( G ) ≤ � 3 n � . 2 These bounds are asymptotically tight. For infinite values of n , there are graphs G such that √ χ g ( G ) + χ g ( G ) = 2 2 n − 1 and � 4 n � χ g ( G ) + χ g ( G ) = − 1. 3 C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 24 / 38

Definitions Norhaus-Gaddum inequalities Result (1) Theorem [Nordhaus and Gaddum, 1956] For any graph G of order n , 2 √ n ≤ χ ( G ) + χ ( G ) ≤ n + 1. These bounds are tight for an infinite number of values of n . Theorem For any graph G of order n , 2 √ n ≤ χ g ( G ) + χ g ( G ) ≤ � 3 n � . 2 These bounds are asymptotically tight. For infinite values of n , there are graphs G such that √ χ g ( G ) + χ g ( G ) = 2 2 n − 1 and � 4 n � χ g ( G ) + χ g ( G ) = − 1. 3 C., Dantas, de Figueiredo, Furtado, Gravier Nordhaus-Gaddum inequalities GAG Workshop 24 / 38