Graph polynomials from simple graph sequences

Transcription

1 Graph polynomials from simple graph sequences Delia Garijo 1 Andrew Goodall 2 Patrice Ossona de Mendez 3 Jarik Nešetřil 2 1 University of Seville, Spain 2 Charles University, Prague, Czech Republic 3 CAMS, CNRS/EHESS, Paris, France KAIST Discrete Math Seminar, 14 April 2014 Dept. of Mathematical Sciences, KAIST, Daedeok Innopolis, Daejeon

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3 Graph polynomials Graph homomorphisms Chromatic polynomial Definition by evaluations at positive integers k N, P(G; k) = #{proper vertex k-colourings of G}. P(G; k) = 1 j V (G) a j (G)k j a j (G) = #{partitions of V (G) into j independent subsets}, P(G; k) = 1 j V (G) ( 1) j V (G) j b j (G)k b j (G) = #{j-subsets of E(G) containing no broken cycle}. uv E(G), P(G; k) = P(G\uv; k) P(G/uv; k)

4 Graph polynomials Graph homomorphisms Independence polynomial Definition by coefficients I (G; x) = b j (G)x j, 1 j V (G) b j (G) = #{independent subsets of V (G) of size j}. v V (G), I (G; x) = I (G v; x) + xi (G N[v]; x) I (L(G); x) = matching polynomial of G (Chudnovsky & Seymour, 2006) K 1,3 i G I (G; x) real roots b 2 j b j 1 b j+1, (implies b 1,..., b V (G) unimodal)

5 Graph polynomials Graph homomorphisms Definition Graphs G, H. f : V (G) V (H) is a homomorphism from G to H if uv E(G) f (u)f (v) E(H). Definition H with adjacency matrix (a s,t ), weight a s,t on st E(H), hom(g, H) = f :V (G) V (H) uv E(G) a f (u),f (v). hom(g, H) = #{homomorphisms from G to H} = #{H-colourings of G} when H simple (a s,t {0, 1}) or multigraph (a s,t N)

6 Graph polynomials Graph homomorphisms The main question Which sequences (H k,l,... ) of simple graphs are such that, for all graphs G, for each k, l, N we have for polynomial p(g)? hom(g, H k,l,... ) = p(g; k, l,... ) Characterizing simple graph sequences (H k,l,... ) with this property gives straightforward characterization for multigraph sequences too (allowing multiple edges & loops).

7 Graph polynomials Graph homomorphisms

8 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example 1 (K k ) hom(g, K k ) = P(G; k) chromatic polynomial

9 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example 2 (K 1 k ) hom(g, Kk 1 ) = k V (G)

10 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example 3 l l l l l l l l l l (K l k ) hom(g, K l k ) = f :V (G) [k] #{uv E(G) f (u)=f (v)} l = k c(g) (l 1) r(g) T (G; l 1+k l 1, l) (Tutte polynomial)

11 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example 4 (K K 1,k) hom(g, K K 1,k) = I (G; k) independence polynomial

12 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example 5 (K k 2 ) = (Q k ) (hypercubes) Proposition (Garijo, G., Nešetřil, 2013+) hom(g, Q k ) = p(g; k, 2 k ) for bivariate polynomial p(g)

13 Examples Strongly polynomial sequences of graphs Counting induced subgraphs Definition (H k ) is strongly polynomial (in k) if G polynomial p(g) such that hom(g, H k ) = p(g; k) for all k N. Since hom(g 1 G 2, H) = hom(g 1, H)hom(G 2, H), suffices to consider connected G. Example (K k ), (K 1 k ). (kk 2) are strongly polynomial (Kk l ) is strongly polynomial (in k, l) (Q k ) not strongly polynomial (but polynomial in k and 2 k ) (C k ), (P k ) not strongly polynomial (but eventually polynomial in k)

14 Examples Strongly polynomial sequences of graphs Counting induced subgraphs Subgraph criterion for strongly polynomial H k simple: hom(g, H k ) = sur V (G, S) S i H k V (S) V (G) = S/ = sur V (G, S) #{induced copies of S in H k } Proposition (de la Harpe & Jaeger 1995) (H k ) is strongly polynomial connected S #{induced subgraphs = S in H k } polynomial in k

15 Examples Strongly polynomial sequences of graphs Counting induced subgraphs Subgraph criterion for strongly polynomial H k simple: hom(g, H k ) = sur V (G, S) S i H k V (S) V (G) = S/ = sur V (G, S) #{induced copies of S in H k } Proposition (de la Harpe & Jaeger 1995) (for each S want this polynomial in k) (H k ) is strongly polynomial connected S #{induced subgraphs = S in H k } polynomial in k

16 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Example: chromatic polynomial hom(g, K k ) = P(G; k) = = as ( k j ) = 0 when j > k V (G). 1 j min{ V (G),k} 1 j V (G) ( ) k sur V (G, K j ) j ( ) k sur V (G, K j ), j

17 What are we looking at? Examples Strongly polynomial sequences of graphs Counting induced subgraphs Eventually polynomial but not strongly polynomial (C k ) hom(g, C k ) = sur V (G, P j ) k + sur V (G, C k ) 1 j min{ V (G),k 1} hom(c 3, C 3 ) = 6, hom(c 3, C k ) = 0 when k = 2 or k 4

18 Constructions Loose threads up until a few months ago...

19 Constructions Loose threads up until a few months ago... Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nešetřil, 2013+) If (H k ) is strongly polynomial and H k simple, then (H k ) (complements), (L(H k )) (line graphs), are also strongly polynomial. Also, (lh k ) is strongly polynomial (in k and l). Proposition (Garijo, G., Nešetřil, 2013+) If (H k ) is strongly polynomial, at most one loop each vertex of H k, then (Hk 0 ) (remove all loops) (Hk 1 ) (add loops to make 1 loop each vertex) are also strongly polynomial. More generally, (Hk l ) is strongly polynomial (in k and l).

20 Constructions Loose threads up until a few months ago... Proposition If (F j ), (H k ) are strongly polynomial, then (F j H k ) (disjoint union) (F j + H k ) (join) (F j H k ) (direct/categorical product) (F j [H k ]) (lexicographic product) are strongly polynomial (in j and k).

21 Constructions Loose threads up until a few months ago... Example Beginning with trivially strongly polynomial sequence (K 1 ), following are also strongly polynomial: multiple: (kk 1 ) = (K k ) complement: (K k ) (chromatic polynomial) loop-addition: (Kk l ) (Tutte polynomial) join: (Kk j 1 + K j l ) (Averbouch Godlin Makowsky polynomial includes Tutte polynomial, satisfies three-term recurrence in \uv, /uv and u v)

22 Constructions Loose threads up until a few months ago... Question Strongly polynomial sequences: (K j + K k ) = (K j,k ) (L(K j,k )) = (K j K k ) (Rook s graph) (F j ), (H k ) strongly polynomial (F j H k ) strongly polynomial?

23 Constructions Loose threads up until a few months ago... Definition Generalized Johnson graph J k,l,d, D {0, 1,..., l} vertices ( [k]) l, edge uv when u v D Johnson graphs D = {k 1} Kneser graphs D = {0} Proposition (de la Harpe & Jaeger, 1995; Garijo, G., Nešetřil, 2013+) For every l, D, sequence (J k,l,d ) is strongly polynomial (in k). Question Can generalized Johnson graphs be generated from simpler sequences by any of the constructions described in de la Harpe & Jaeger (1995) and Garijo, Goodall & Nešetřil (2013+)?

24 Constructions Loose threads up until a few months ago...

25 Recap Relational structures Example interpretations Everything? Simple graph sequence (H k ) strongly polynomial iff G polynomial p(g) k N hom(g, H k ) = p(g; k) F polynomial q(f ) k N ind(f, H k ) = q(f ; k) Unary operations and binary operations such that if simple graph sequences (F j ) and (H k ) are strongly polynomial then ( H k ) is strongly polynomial (e.g. complement, line graph) (F j H k ) is strongly polynomial in j, k (e.g. join, lexicographic product)

26 Recap Relational structures Example interpretations Everything? Satisfaction sets Quantifier-free formula φ with n free variables (φ QF n ) with symbols from relational structure H with domain V (H). Satisfaction set φ(h) = {(v 1,..., v n ) V (H) n : H = φ}. e.g. for graph structure H (symmetric binary relation x y interpreted as x adjacent to y), and given graph G on n vertices, φ = φ G = (v i v j ) ij E(G) φ G (H) = {(v 1,..., v n ) : i v i is a homomorphism G H} φ G (H) = hom(g, H).

27 Recap Relational structures Example interpretations Everything? Strongly polynomial sequences of structures Definition Sequence (H k ) of relational structures strongly polynomial iff φ QF polynomial r(φ) k N φ(h k ) = r(φ; k) Lemma Equivalently, G polynomial p(g) k N F polynomial q(f) k N hom(g, H k ) = p(g; k), or ind(f, H k ) = q(f; k). Transitive tournaments ( T k ) strongly polynomial sequence of digraphs (e.g. count induced substructures).

28 Recap Relational structures Example interpretations Everything? Graphical QF interpretation schemes Lemma There is I : Relational σ-structures A Graphs H Ĩ : φ QF(Graphs) Ĩ (φ) QF(σ-structures) such that φ(i (A)) = Ĩ (φ)(a) In particular, (A k ) strongly polynomial (H k ) = (I (A k )) strongly polynomial.

29 Recap Relational structures Example interpretations Everything? From graphs to graphs All previous constructions (complementation, line graph, disjoint union, join, direct product,...) special cases of interpretation schemes I from Marked Graphs (added unary relations) to Graphs. Cartesian product and other more complicated graph products are special kinds of such interpretation schemes too.

30 What are we looking at? Recap Relational structures Example interpretations Everything? Generalized Johnson graphs (J k,l,d ) arise as QF interpretations of transitive tournaments T k Half-graphs are QF interpretations of a transitive tournament together with marks (unary relations used to specfiy upper + lower vertices) and so form a strongly polynomial sequence. upper lower join upper vertices to lower vertices to the right

31 Recap Relational structures Example interpretations Everything? Intersection graphs of chords of a k-gon form a strongly polynomial sequence b a a a b a b c c e d b e d (a) Square (b) Pentagon d e f g h a b c i d a e c b g i f h a h g n f m i b e l j c k d g f a n m l e h k b i j d c (c) Hexagon (d) Heptagon

32 Recap Relational structures Example interpretations Everything? Conjecture All strongly polynomial sequences of graphs (H k ) can be obtained by QF interpretation of a basic sequence (disjoint union of marked transitive tournaments of size polynomial in k).

33 Recap Relational structures Example interpretations Everything?

34 Paley graphs Further problems Prime power q = p d 1 (mod 4) Paley graph P q =Cayley(F q, non-zero squares), Quasi-random graphs: hom(g, P q )/hom(g, G q, 1 ) 1 as q. 2 Proposition (Corollary to result of de la Harpe & Jaeger, 1995) hom(g, P q ) is polynomial in q for series-parallel G. e.g. hom(k 3, P q ) = q(q 1)(q 5) 8 Prime q 1 (mod 4), q = 4x 2 + y 2, [Evans, Pulham, Sheehan, 1981]: q(q 1) ( hom(k 4, P q ) = (q 9) 2 4x 2) 1536 Is hom(g, P q ) polynomial in q and x for all graphs G? Theorem (G., Nešetřil, Ossona de Mendez, 2014+) If (H k ) is strongly polynomial then there are only finitely many terms belonging to a quasi-random sequence of graphs.

35 Paley graphs Further problems When is (Cayley(A k, B k )) polynomial in A k, B k, where B k = B k A k? e.g. For D N, sequence ( Cayley(Z k, ±D) ) is polynomial iff D is finite or cofinite. (de la Harpe & Jaeger, 1995) Can (H k ) be verified to be strongly polynomial by testing hom(g, H k ) for G only in a restricted class of graphs? (yes, for connected graphs but for a smaller class?) Which graph polynomials defined by strongly polynomial sequences of graphs satisfy a reduction formula (size-decreasing recurrence) like the chromatic polynomial and independence polynomial? Develop similar theory for hom(h k, G) (e.g. hom(c k, G) = λ k, λ eigenvalues of G, determines characteristic polynomial of G by its roots).

36 Paley graphs Further problems

37 Paley graphs Further problems Three papers P. de la Harpe and F. Jaeger, Chromatic invariants for finite graphs: theme and polynomial variations, Lin. Algebra Appl (1995), Defining graphs invariants from counting graph homomorphisms. Examples. Basic constructions. D. Garijo, A. Goodall, J. Nešetřil, Polynomial graph invariants from homomorphism numbers. 40pp. arxiv: [math.co] Further examples. New construction using tree representations of graphs. A. Goodall, J. Nešetřil, P. Ossona de Mendez, Strongly polynomial sequences as interpretation of trivial structures. 17pp. Preprint. General relational structures: counting satisfying assignments for quantifier-free formulas. Building new polynomial invariants by interpretation of trivial sequences of marked tournaments.