Clique colouring of random graphs

Transcription

1 Clique colouring of random graphs Colin McDiarmid Oxford University Scottish Combinatorics Meeting Glasgow, April 2016 Colin McDiarmid (Oxford) Clique colourins 1 / 16

2 joint work This is joint work with Nikola Yolov (Oxford University), concerning random perfect graphs, and with Dieter Mitsche (Université de Nice Sophia-Antipolis) and Pawe l Pra lat (Ryerson University, Toronto), concerning random geometric graphs and Erdős-Rényi (or binomial) random graphs. Colin McDiarmid (Oxford) Clique colourins 2 / 16

3 Clique colourings 1 Given a graph G, a clique is a subset S of the vertex set such that any pair of vertices in S is connected by an edge; and a clique is maximal if it is not a proper subset of another clique. A clique colouring of G is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number χ c (G). (If G has no edges we take χ c (G) to be 1.) Colin McDiarmid (Oxford) Clique colourins 3 / 16

4 Clique colourings 2 Repeat: A clique colouring of G is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices); and the least possible number of colours is the clique chromatic number χ c (G). χ c (G) is at most the chromatic number χ(g), but it is possible for χ c (G) to be much smaller. For any n 2 we have χ(k n ) = n but χ c (K n ) = 2. Adding edges can decrease or increase χ c (G). If G is triangle-free then χ c (G) = χ(g). Planar graphs G satisfy χ c (G) 3, Mohar and Skrekovski Colin McDiarmid (Oxford) Clique colourins 4 / 16

5 Clique colourings 2 Repeat: A clique colouring of G is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices); and the least possible number of colours is the clique chromatic number χ c (G). χ c (G) is at most the chromatic number χ(g), but it is possible for χ c (G) to be much smaller. For any n 2 we have χ(k n ) = n but χ c (K n ) = 2. Adding edges can decrease or increase χ c (G). If G is triangle-free then χ c (G) = χ(g). Planar graphs G satisfy χ c (G) 3, Mohar and Skrekovski Colin McDiarmid (Oxford) Clique colourins 4 / 16

6 Clique colouring and perfect graphs 1 It is easy to see that χ c 2 for bipartite graphs, co-bipartite graphs, comparability graphs,.. For many classes of graphs, in particular for many subclasses of perfect graphs, it is known that χ c (G) 3. For example this holds for co-comparability graphs (Duffus, Kierstead and Trotter 1991) and generalised split graphs (Bacsó, Gravier, Gyárfás, Preissmann and Sebő 2004). It was asked in 1991 if χ c (G) 3 for all perfect graphs. This year Charbit, Penev, Thomassé and Trotignon showed to the contrary that χ c (G) is unbounded on perfect graphs. Colin McDiarmid (Oxford) Clique colourins 5 / 16

7 Clique colouring and perfect graphs 1 It is easy to see that χ c 2 for bipartite graphs, co-bipartite graphs, comparability graphs,.. For many classes of graphs, in particular for many subclasses of perfect graphs, it is known that χ c (G) 3. For example this holds for co-comparability graphs (Duffus, Kierstead and Trotter 1991) and generalised split graphs (Bacsó, Gravier, Gyárfás, Preissmann and Sebő 2004). It was asked in 1991 if χ c (G) 3 for all perfect graphs. This year Charbit, Penev, Thomassé and Trotignon showed to the contrary that χ c (G) is unbounded on perfect graphs. Colin McDiarmid (Oxford) Clique colourins 5 / 16

8 Clique colouring and perfect graphs 1 It is easy to see that χ c 2 for bipartite graphs, co-bipartite graphs, comparability graphs,.. For many classes of graphs, in particular for many subclasses of perfect graphs, it is known that χ c (G) 3. For example this holds for co-comparability graphs (Duffus, Kierstead and Trotter 1991) and generalised split graphs (Bacsó, Gravier, Gyárfás, Preissmann and Sebő 2004). It was asked in 1991 if χ c (G) 3 for all perfect graphs. This year Charbit, Penev, Thomassé and Trotignon showed to the contrary that χ c (G) is unbounded on perfect graphs. Colin McDiarmid (Oxford) Clique colourins 5 / 16

9 Clique colouring and perfect graphs 2 A graph is unipolar if the vertex set can be partitioned into a clique C 0 and a disjoint union of cliques C 1, C 2,... (with no edges between the C i for i 1). For example, the graph obtained from C 9 by adding edges to form an evenly spaced triangle is unipolar, with χ c = 3. A graph G is a generalised split graph if it or its complement G is unipolar. Colin McDiarmid (Oxford) Clique colourins 6 / 16

10 Clique colouring and perfect graphs 3 Prömel and Steger showed in 1992 that almost all perfect graphs are generalised split graphs. Bacsó, Gravier, Gyárfás, Preissmann and Sebő 2004 showed that all generalised split graphs are 3-clique-colourable, and deduced that almost all perfect graphs are 3-clique-colourable. Theorem (McD and Yolov 2016+) Almost all perfect graphs are 2-clique-colourable. Colin McDiarmid (Oxford) Clique colourins 7 / 16

11 Clique colouring and perfect graphs 3 Prömel and Steger showed in 1992 that almost all perfect graphs are generalised split graphs. Bacsó, Gravier, Gyárfás, Preissmann and Sebő 2004 showed that all generalised split graphs are 3-clique-colourable, and deduced that almost all perfect graphs are 3-clique-colourable. Theorem (McD and Yolov 2016+) Almost all perfect graphs are 2-clique-colourable. Colin McDiarmid (Oxford) Clique colourins 7 / 16

12 Geometric graphs Given n points x 1,..., x n in R 2 and given a threshold r > 0, the corresponding (Euclidean) geometric graph has vertex set {v 1,..., v n }, and for i j, vertices v i and v j are adjacent when the Euclidean distance d(x i, x j ) is at most r. We call a graph G geometric if there are points x j and r > 0 realising G as above. By rescaling by a factor 1/r we may assume wlog that r = 1. A geometric graph is also called a unit disk graph. Colin McDiarmid (Oxford) Clique colourins 8 / 16

13 Geometric graphs and clique colouring 1 Theorem Every geometric graph G has χ c (G) 9. To prove this, partition the plane into horizontal strips S n = R [ny, (n + 1)y) of height y, where 1 2 y 3/2. Since y 3/2 the graph induced on a strip is a co-comparability graph, and so has χ c 3. Since y 1 2, we can use the same 3 colours on every third strip, so we need at most 9 in total. Colin McDiarmid (Oxford) Clique colourins 9 / 16

14 Geometric graphs and clique colouring 1 Theorem Every geometric graph G has χ c (G) 9. To prove this, partition the plane into horizontal strips S n = R [ny, (n + 1)y) of height y, where 1 2 y 3/2. Since y 3/2 the graph induced on a strip is a co-comparability graph, and so has χ c 3. Since y 1 2, we can use the same 3 colours on every third strip, so we need at most 9 in total. Colin McDiarmid (Oxford) Clique colourins 9 / 16

15 Geometric graphs and clique colouring 1 Theorem Every geometric graph G has χ c (G) 9. To prove this, partition the plane into horizontal strips S n = R [ny, (n + 1)y) of height y, where 1 2 y 3/2. Since y 3/2 the graph induced on a strip is a co-comparability graph, and so has χ c 3. Since y 1 2, we can use the same 3 colours on every third strip, so we need at most 9 in total. Colin McDiarmid (Oxford) Clique colourins 9 / 16

16 Geometric graphs and clique colouring 1 Theorem Every geometric graph G has χ c (G) 9. To prove this, partition the plane into horizontal strips S n = R [ny, (n + 1)y) of height y, where 1 2 y 3/2. Since y 3/2 the graph induced on a strip is a co-comparability graph, and so has χ c 3. Since y 1 2, we can use the same 3 colours on every third strip, so we need at most 9 in total. Colin McDiarmid (Oxford) Clique colourins 9 / 16

17 Geometric graphs and clique colouring 2 Let χ max c (R 2 ) denote the maximum value of χ c (G) over geometric graphs G in R 2. Then 3 χ max c (R 2 ) 9. Can we improve these bounds? Colin McDiarmid (Oxford) Clique colourins 10 / 16

18 Random geometric graphs Let r > 0, and consider n random points X 1,..., X n independently and uniformly distributed in the square [ n/2, n/2] 2. We call the graph formed from such points the random geometric graph G(n, r). Colin McDiarmid (Oxford) Clique colourins 11 / 16

19 Random geometric graphs and clique colouring 1 Theorem For the random geometric graph G G(n, r) in the plane, with high probability = 1 if nr 0 = 2 if nr and nr 8 0 χ c (G) 3 if nr 8 / log n and r 0.46 log n = 2 if r 9.3 log n Thus as r increases from 0 we have whp the following rough picture: χ c (G) is 1 up to about n 1, then 2 up to about n 1/8, then at least 3 (and at most χ max c (R 2 ) 9) up to about log n (roughly the connectivity threshold), when it drops back to 2 and remains there. Colin McDiarmid (Oxford) Clique colourins 12 / 16

20 Random geometric graphs and clique colouring 2 We do not know the full story around the transition points. However, we can say more within the interval where χ c (G) 3; namely that within a subinterval χ c (G) is whp as large as is possible for a geometric graph. Theorem There exists ε > 0 such that, for G G(n, r) with n ε r ε log n, whp we have χ c (G) = χ max c (R 2 ). Colin McDiarmid (Oxford) Clique colourins 13 / 16

21 Random geometric graphs and clique colouring 2 We do not know the full story around the transition points. However, we can say more within the interval where χ c (G) 3; namely that within a subinterval χ c (G) is whp as large as is possible for a geometric graph. Theorem There exists ε > 0 such that, for G G(n, r) with n ε r ε log n, whp we have χ c (G) = χ max c (R 2 ). Colin McDiarmid (Oxford) Clique colourins 13 / 16

22 G(n, p) and clique colouring 1 Let x 0 < x < 1/2 f (x) = 3x/2 1/2 1/2 x < 3/5 1 x 3/5 x < 1. Colin McDiarmid (Oxford) Clique colourins 14 / 16

23 G(n, p) and clique colouring 2 Theorem Let G G(n, p) for some p = p(n), where np = n x+o(1) for some x (0, 1/2) (1/2, 1) (so x log np/ log n). Then whp χ c (G) = n f (x)+o(1). (The proof of the lower bound for x ( 1 2, 1) is not yet fully written down.) Colin McDiarmid (Oxford) Clique colourins 15 / 16

24 The end Thanks for your attention! Colin McDiarmid (Oxford) Clique colourins 16 / 16