Extremal Restraints for Graph Colourings

Transcription

1 Extremal Restraints for Graph Colourings Aysel Erey Dalhousie University CanaDAM 2015 University of Saskatchewan, Saskatoon, June 1 (Joint work with Jason Brown)

2 Definition A proper k-colouring of G is a function c : V (G) {1,..., k} such that c(u) c(v) for every edge uv of G. Definition The chromatic polynomial π(g, k) is the polynomial whose evaluation at a positive integer k counts the number of proper k-colourings of G.

3 A little bit history... Chromatic polynomial was introduced by Birkhoff Four Colour Theorem (Appel-Haken 1976) Every planar graph is 4-colourable. (In other words, 4 cannot be a root of the chromatic polynomial of a planar graph.) Picture Source:

4 Chromatic polynomial encodes combinatorial information Multiplicity of the root 0 = number of components Multiplicity of the root 1 = number of blocks π(g, 1) = number of acyclic orientations Connections to other polynomials Tutte polynomial Rank polynomial Characteristic polynomials of matroids Potts model partition function The complex roots play an important role in statistical mechanics

5 Restrained colourings of graphs Restrained Coloring A restraint r on a graph G is a function which assigns each vertex of G a finite set of forbidden colours r(v). A proper x-coloring c is called an x-coloring permitted by r if for each vertex v, c(v) is not from r(v). Restrained colourings Arise naturally as graph is sequentially coloured; Are of use in the construction of critical graphs; Have applications to scheduling and timetabling.

6 Definition The restrained chromatic polynomial π r (G, x) of G counts the number of proper x-colourings permitted by restraint r. Theorem π r (G, x) is a polynomial function of x when x is large enough!

7 Restrained Chromatic Polynomial and Basic Properties The restrained chromatic polynomial generalizes the chromatic polynomial π r (G, x) = π(g, x) when r(u) = for every vertex u, is monic polynomial of degree n, has integer coefficients, its coefficients alternate in sign, satisfies π r (G, x) = π r (G 1, x) π r (G k, x) where G 1,... G k are the connected components of G, and its constant term need not be 0.

8 Edge Addition-Contraction Formula for the Chromatic Polynomial For every nonedge e = uv of G, π(g, x) = π(g + e, x) + π(g e, x) u v u v w Figure: From left to right: G, G + e, G e

9 u v u v w Figure: From left to right: G, G + e, G e Edge Addition-Contraction Formula for the Restrained Chromatic Polynomial For every nonedge e = uv of G, π r (G, x) = π r (G + e, x) + π re (G e, x) { r(a) if a w r e (a) = r(u) r(v) if a = w

10 Definition Two restraints r and r on a graph G are said to be equivalent, denoted r r, if there exists a graph automorphism φ of G and a bijective function f : u V (G) r(u) u V (G) r (u) such that f (r(u)) = r (φ(u)) for every vertex u of G. If r and r are not equivalent then we call them nonequivalent restraints and write r r.

11 Example Figure: From left to right: r 1, r 2, r 3, r 4 r 1 r 2, r 3 r 4, r 1 r 3 If r r then π r (G, x) = π r (G, x) for large enough x.

12 r = [r(v 1 ), r(v 2 ), r(v 3 ), r(v 4 )] v 1 v 2 v 4 v 3

13 Definition A restraint r on G is called a k-restraint if r(u) = k for every vertex u of G. Example All nonequivalent 1-restraints on C 4 r 1 = [{1}, {1}, {1}, {1}] r 2 = [{1}, {1}, {1}, {2}] r 3 = [{1}, {1}, {2}, {2}] r 4 = [{1}, {2}, {1}, {2}] r 5 = [{1}, {1}, {2}, {3}] r 6 = [{1}, {2}, {1}, {3}] r 7 = [{1}, {2}, {3}, {4}]

14 Problem Among all k-restraints r on G such that r(v (G)) [nk] what k-restraint on G permits the largest/smallest number of x-colourings for all large enough x? Remark Such extremal restraints always exist!

15 π ri (C 4, x) All 1-restraints on C 4 r 1 = [{1}, {1}, {1}, {1}] (min) r 2 = [{1}, {1}, {1}, {2}] r 3 = [{1}, {1}, {2}, {2}] r 4 = [{1}, {2}, {1}, {2}] (max) r 5 = [{1}, {1}, {2}, {3}] r 6 = [{1}, {2}, {1}, {3}] r 7 = [{1}, {2}, {3}, {4}]

16 Definition Let R max (G, k) (resp. R min (G, k)) be the set of k-restraints on G permitting the maximum (resp. minimum) number of x-colourings for all large enough x.

17 Restraints permitting the smallest number of colourings Definition A restraint r is called a constant restraint if r(u) = r(v) for every pair of vertices u and v of G. Theorem (J.B A.E) Let G be a connected graph. Then r R min (G, k) r is a constant restraint.

18 Kostochka and Sidorenko (1990) If a chordal graph G has a list of at least l available colours at every vertex, then for every natural number l, the number of list colourings π(g, l). There exist graphs G such that for some natural number l, the number of list colourings < π(g, l). Donner (1992) J. Graph Theory Thomassen (2009) J. Combin. Theory Ser. B Let G be any graph then the number of list colourings π(g, l) provided l is sufficiently large.

19 We determined the restraint which belongs to R min (G, k). Such restraint is unique (up to equivalence of restraints) and the same for all graphs. How to determine restraint(s) which belong(s) to R max (G, k)? Such restraints are more difficult to determine and they change from graph to graph.

20 Three necessary conditions for a restraint to be in R max (G, k) for all graphs G. Determine the restraint in R max (G, k) when G is a complete graph bipartite graph.

21 First necessary condition for a restraint to be in R max (G, k) Definition A restraint r on G is called a proper restraint if r(u) r(v) = for every edge uv E(G). Theorem (J.B A.E) Let G be any graph. If r R max (G, k) then r is a proper restraint.

22 All 1-restraints on C 4 r 1 = [{1}, {1}, {1}, {1}] r 2 = [{1}, {1}, {1}, {2}] r 3 = [{1}, {1}, {2}, {2}] r 4 = [{1}, {2}, {1}, {2}] r 5 = [{1}, {1}, {2}, {3}] r 6 = [{1}, {2}, {1}, {3}] r 7 = [{1}, {2}, {3}, {4}] All proper 1-restraints on C 4 r 4 = [{1}, {2}, {1}, {2}] r 6 = [{1}, {2}, {1}, {3}] r 7 = [{1}, {2}, {3}, {4}]

23 Corollary (J.B A.E) Let r be a k-restraint on a complete graph K n such that r (u) r (v) = for all vertices u v. Then for every k-restraint r r on G π r (G, x) < π r (G, x) for all large enough x.

24 π ri (K 5, x) All 1-restraints on K 5 r 1 = [{1}, {1}, {1}, {1}, {1}] (min) r 2 = [{1}, {1}, {1}, {1}, {2}] r 3 = [{1}, {1}, {1}, {2}, {2}] r 4 = [{1}, {1}, {1}, {2}, {3}] r 5 = [{1}, {1}, {2}, {3}, {4}] r 6 = [{1}, {1}, {2}, {2}, {3}] r 7 = [{1}, {2}, {3}, {4}, {5}] (max)

25 Second necessary condition for a restraint r to be in R max (G, k) Theorem (J.B A.E) Let G be any graph. If r R max (G, k) then r maximizes the expression r(v) r(w) u V (G) v,w N G (u) v w among all proper k-restraints r. u v w

26 Example Let G = C 4 First condition: All proper 1-restraints on C 4 r 4 = [{1}, {2}, {1}, {2}] r 6 = [{1}, {2}, {1}, {3}] r 7 = [{1}, {2}, {3}, {4}] Second condition: u V (G) v,w N G (u) v w r(v) r(w) = 4 if r = r 4 2 if r = r 6 0 if r = r 7 Conclusion: R max (G, k) = {r 4 }

27 Bipartite Graphs Definition Let V (G) = A B be the partition of a bipartite graph G. Then a restraint function is called an alternating restraint, denoted r alt, if r alt is constant on both A and B individually but r alt (A) r alt (B). Theorem (J.B A.E) Let G be a connected bipartite graph. Then r R max (G, k) r r alt.

28 Theorem (J.B A.E) Let G be any graph. If r R max (G, k) then r is a proper restraint and it maximizes the expression u V (G) v,w N G (u) v w among all proper k-restraints r. r(v) r(w) These two necessary conditions are sufficient to determine R max (G, k) when G is bipartite. Are they sufficient to determine R max (G, k) for a general graph? No.

29 Example There are two nonequivalent proper restraints achieving the maximum value of r(v) r(w). u V (G) v,w N G (u) v w

30 Third necessary condition on r to be in R max (G, k) (A.E J.B) Let G be a graph with girth at least 5 and r R max (G, k). Then r maximizes the expression r(v) r(w) r(t) u V (G) v,w,t N G (u) v,w,t distinct uv E(G) u N G (u)\{v} v N G (v)\{u} r(u ) r(v ) among all proper k-restraints r satisfying the second condition. u v u u v v w t

31 Figure: Two nonequivalent proper restraints r 1 (left) and r 2 (right). uv E(G) u N G (u)\{v} v N G (v)\{u} { r(u ) r(v 2 if r = r1 ) = 1 if r = r 2 Thus, R max (G, k) = {r 2 }

32 Problem Are the extremal restraints always unique (up to equivalence)? R min (G, k) consists of a unique restraint for every graph G R max (G, k) consists of a unique restraint when G is a complete graph bipartite graph

33 It is possible that a graph has the same restrained chromatic polynomial for two nonequivalent restraints. Example For two nonequivalent restraints r = [{1}, {2}, {2}, {1}] and r = [{1}, {2}, {3}, {3}] on P 4, π r (P 4, x) = π r (P 4, x) = x 4 7 x x 2 28 x + 16 for all large enough x.

34 Definition Let R max (G, k) (resp. R min (G, k)) be the set of k-restraints on G permitting the maximum (resp. minimum) number of x-colourings for all large enough x. Question How large x needs to be?

35 Thomassen (2009) J. Combin. Theory Ser. B Let G be any graph then the number of list colourings π(g, l) provided l n 10. Corollary If r r c is a k-restraint on G then π rc (G, x) π r (G, x) for every x n 10 + nk.

36 Theorem (J.B A.E) Let r be a k-restraint on a complete graph K n such that r (u) r (v) = for all vertices u v. Then for every k-restraint r r on G π r (G, x) < π r (G, x) for all x nk. Theorem (J.B A.E) Let G be a tree and r alt be an alternating k-restraint on G. Then, for any k-restraint r r alt, for all x nk. π r (T, x) < π ralt (T, x)

37 The End Thank you!