Minor Monotone Floors and Ceilings of Graph Parameters

Transcription

1 Minor Monotone Floors and Ceilings of Graph Parameters Thomas Milligan Department of Mathematics and Statistics University of Central Oklahoma 13 July, SIAM Annual Meeting - Minneapolis, MN (Work with Xander Rudelis) Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

2 Background - Colin De Verdiére Graph Parameter First, we consider the set of real symmetric matrices S n and simple undirected graphs with no loops and no multiple edges. For n n real symmetric matrix A, then G(A) has vertices {1,..., n} and as edges, {i, j} exactly when a i,j 0 with i j. We define S(G) = {A S n G(A) = G} and S A = S(G(A)). Colin De Verdiére parameter The Colin De Verdiére parameter µ(g) is the maximum nullity over all A S n that satisfy: A S(G) A satisfies the Strong Arnold Property (SAP) For all i j, a i,j 0 A has exactly one negative eigenvalue (including multiplicities) Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

3 Background - Colin De Verdiére Graph Parameter First, we consider the set of real symmetric matrices S n and simple undirected graphs with no loops and no multiple edges. For n n real symmetric matrix A, then G(A) has vertices {1,..., n} and as edges, {i, j} exactly when a i,j 0 with i j. We define S(G) = {A S n G(A) = G} and S A = S(G(A)). Colin De Verdiére parameter The Colin De Verdiére parameter µ(g) is the maximum nullity over all A S n that satisfy: A S(G) A satisfies the Strong Arnold Property (SAP) For all i j, a i,j 0 A has exactly one negative eigenvalue (including multiplicities) Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

4 Strong Arnold Property Strong Arnold Property (SAP) A real symmetric matrix A is said to have the SAP provided the only real symmetrix matrix X satisfying AX = 0, A X = 0 and I X = 0 is the zero matrix. Other Colin De Verdiére type parameters: ν(g) and ξ(g). Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

5 Strong Arnold Property Strong Arnold Property (SAP) A real symmetric matrix A is said to have the SAP provided the only real symmetrix matrix X satisfying AX = 0, A X = 0 and I X = 0 is the zero matrix. Other Colin De Verdiére type parameters: ν(g) and ξ(g). Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

6 Strong Arnold Property Strong Arnold Property (SAP) A real symmetric matrix A is said to have the SAP provided the only real symmetrix matrix X satisfying AX = 0, A X = 0 and I X = 0 is the zero matrix. Other Colin De Verdiére type parameters: ν(g) and ξ(g). Interesting property of Colin De Verdiére type parameters: They are minor monotone. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

7 Strong Arnold Property Strong Arnold Property (SAP) A real symmetric matrix A is said to have the SAP provided the only real symmetrix matrix X satisfying AX = 0, A X = 0 and I X = 0 is the zero matrix. Other Colin De Verdiére type parameters: ν(g) and ξ(g). Interesting property of Colin De Verdiére type parameters: They are minor monotone. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

8 Graph Minors We say that X is a minor of Y, written X Y if X can be obtained from Y through a series of subgraphs and edge contractions. Example: Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

9 Minor Monotone We say a real-valued graph parameter p is minor monotone if X Y implies p(x ) p(y ). Most graph parameters are not minor monotone. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

10 Minor Monotone We say a real-valued graph parameter p is minor monotone if X Y implies p(x ) p(y ). Most graph parameters are not minor monotone. For example: X: Y: Note that while X Y, we see that the minimum degree is not minor monotone, as δ(x ) = 2 and δ(y ) = 0. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

11 Minor Monotone We say a real-valued graph parameter p is minor monotone if X Y implies p(x ) p(y ). Most graph parameters are not minor monotone. For example: X: Y: Note that while X Y, we see that the minimum degree is not minor monotone, as δ(x ) = 2 and δ(y ) = 0. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

12 Motivation Finite Sets of Forbidden Minors One of the fundamental results of Robertson and Seymour says that given minor monotone graph parameter p and real value k, the set of graphs such that p(g) k can be characterized by excluding a finite set of forbidden minors. It has been shown that: µ(g) 1 if and only if G is a disjoint union of paths. µ(g) 2 if and only if G is outerplanar. µ(g) 3 if and only if G is planar. µ(g) 4 if and only if G is linklessly embeddable. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

13 Other Minor monotone graph parameters Other examples of minor monotone graph parameters: Tree-width Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

14 Other Minor monotone graph parameters Other examples of minor monotone graph parameters: Tree-width Graph genus Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

15 Other Minor monotone graph parameters Other examples of minor monotone graph parameters: Tree-width Graph genus Minor-crossing number Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

16 Other Minor monotone graph parameters Other examples of minor monotone graph parameters: Tree-width Graph genus Minor-crossing number Definition [Bokal, Fijavž, Mohar:2005] For given graph G, the minor crossing number is the minimum crossing number over all graphs that contain G as a minor, that is: mcr(g) = min{cr(h) G H}. mcr(g) 1 2 (m 3n + 6) mcr(k n ) 1 4 (n 3)(n 4) for n 3 mcr(k n ) 1 2 (n 5)2 + 3 for n (m 2)(n 2) mcr(k m,n) (m 3)(n 3) + 5 for n m 3 Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

17 Other Minor monotone graph parameters Other examples of minor monotone graph parameters: Tree-width Graph genus Minor-crossing number Definition [Bokal, Fijavž, Mohar:2005] For given graph G, the minor crossing number is the minimum crossing number over all graphs that contain G as a minor, that is: mcr(g) = min{cr(h) G H}. mcr(g) 1 2 (m 3n + 6) mcr(k n ) 1 4 (n 3)(n 4) for n 3 mcr(k n ) 1 2 (n 5)2 + 3 for n (m 2)(n 2) mcr(k m,n) (m 3)(n 3) + 5 for n m 3 Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

18 Definition Minor Monotone Floor Let p be a graph parameter with well-ordered range. The minor monotone floor of p is p (G) = min{p(h) : G H}. Minor Monotone Ceiling Let p be a graph parameter with well-ordered range. The minor monotone ceiling of p is p (G) = max{p(h) : H G}. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

19 Trivial Examples For graph G on n vertices and m edges, the following can be shown: Minor Monotone Floor: b 0 (G) = 1 δ (G) = 0 κ (G) = 0 ω (G) = min{2, ω(g)} χ (G) = min{2, χ(g)} (G) = min{3, (G)} ε (G) = 0 α (G) = 1 Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

20 Trivial Examples For graph G on n vertices and m edges, the following can be shown: Minor Monotone Floor: b 0 (G) = 1 δ (G) = 0 κ (G) = 0 ε (G) = 0 α (G) = 1 ω (G) = min{2, ω(g)} χ (G) = min{2, χ(g)} (G) = min{3, (G)} Minor Monotone Ceilings: b 0 (G) = n α (G) = n Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

21 Trivial Examples For graph G on n vertices and m edges, the following can be shown: Minor Monotone Floor: b 0 (G) = 1 δ (G) = 0 κ (G) = 0 ε (G) = 0 α (G) = 1 ω (G) = min{2, ω(g)} χ (G) = min{2, χ(g)} (G) = min{3, (G)} Minor Monotone Ceilings: b 0 (G) = n α (G) = n Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

22 Previously Known Results Known Results: ξ(g) M (G) ν(g) M + (G) κ (G) δ (G) κ (G) ν(g) h(g) 1 κ (G) For G K c n, ppw(g) = Z (G) = CCR Z (G) pw(g) = CCR Z l (G) = Z l (G) la(g) = tstw(g) = CCR Z + (G) = Z + (G). Results from [Barioli, Barrett, Fallat, Hall, Hogben, Shader, van den Driessche, van der Holst: 2012] Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

23 Previously Known Results Known Results: ξ(g) M (G) ν(g) M + (G) κ (G) δ (G) κ (G) ν(g) h(g) 1 κ (G) For G K c n, ppw(g) = Z (G) = CCR Z (G) pw(g) = CCR Z l (G) = Z l (G) la(g) = tstw(g) = CCR Z + (G) = Z + (G). Results from [Barioli, Barrett, Fallat, Hall, Hogben, Shader, van den Driessche, van der Holst: 2012] Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

24 New Results Theorem Let G be a simple connected graph on n vertices. Then Z (G) n α(g) + 1. Conjecture Let G be a simple connected graph on n vertices. Then Z(G) (G). Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

25 New Results Theorem Let G be a simple connected graph on n vertices. Then Z (G) n α(g) + 1. Conjecture Let G be a simple connected graph on n vertices. Then Z(G) (G). Theorem Let G be a simple connected graph on n vertices. Suppose T is a spanning tree with L leaves. Then L (G) Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

26 New Results Theorem Let G be a simple connected graph on n vertices. Then Z (G) n α(g) + 1. Conjecture Let G be a simple connected graph on n vertices. Then Z(G) (G). Theorem Let G be a simple connected graph on n vertices. Suppose T is a spanning tree with L leaves. Then L (G) Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

27 Minors and Complete Graphs A matching M in graph G is a set of pairwise nonadjacent edges. The matching number ν m (G) of a graph G is the number of edges in a maximum matching. Let A = (a 1, a 2,..., a k ) with terms ordered nondecreasingly. We use K A to denote the k partite complete graph K a1,a 2,...,a k. The following three operations on k partite complete graphs yield k partite complete graphs as minors. Partition-Edge Deletion Merge two partitions by deleting all edges between their vertices: K B K A if B = (A\(a x, a y )) (a x + a y ) where x y. Example: K 4,6 K 2,4,4. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

28 Minors and Complete Graphs (cont.) Vertex Deletion Partitions may shrink by removal of vertices from the partition: Example: K 2,3,4,5 K 2,4,4,6. Edge Contraction K B K A if b i a i for all 1 b i B. Create a new partition by contracting an edge. A new partition is created provided the incident vertices are in partitions with at least two vertices. K B K A if B = (A\(a x, a y )) (1, a x 1, a y 1 ). Example: K 1,1,1,1,1 K 1,1,2,2 K 1,3,3 K 4,4. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

29 It can be shown that any complete k partite minor of a complete k partite graph can be obtained using some combination of these operations. From this follows: Theorem For any complete multipartite graph K n1,n 2,...,n k, ω (K n1,n 2,...,n k ) = χ(k n1,n 2,...,n k ) + ν m (K n1 1,n 2 1,...,n k 1) Corollary For any complete multipartite graph K n1,n 2,...,n k, k 1 ω (K n1,n 2,...,n k ) = k + min{ (n i 1), 1 2 i=1 k (n i 1) } i=1 Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

30 It can be shown that any complete k partite minor of a complete k partite graph can be obtained using some combination of these operations. From this follows: Theorem For any complete multipartite graph K n1,n 2,...,n k, ω (K n1,n 2,...,n k ) = χ(k n1,n 2,...,n k ) + ν m (K n1 1,n 2 1,...,n k 1) Corollary For any complete multipartite graph K n1,n 2,...,n k, k 1 ω (K n1,n 2,...,n k ) = k + min{ (n i 1), 1 2 i=1 k (n i 1) } i=1 Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

31 Hadwiger Conjecture Note that the Hadwiger number h(g) of a graph G is defined to be the number of vertices h in the largest complete graph K h that is a minor of G. Using our notation, this is equivalent to saying ω (G) = h(g) χ(g). Theorem Let G = K n1,...,n k be a complete k-partite graph. Let G be a graph obtained from G by deleting any m edges where m ω (G) χ(g) = ν m (K n1 1,n 2 1,...,n k 1). Then G satisfies the Hadwiger Conjecture. Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16

32 Thank you for listening! Thomas Milligan (UCO) Minor Monotone Floors and Ceilings July / 16