Incidence Posets and Cover Graphs

Transcription

1 AMS Southeastern Sectional Meeting, Louisville, Kentucky Incidence Posets and Cover Graphs William T. Trotter, Ruidong Wang Georgia Institute of Technology October 5, 2013 William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 1/18

2 Order diagrams and cover graphs f e a e c d f d a b c b Order Diagram Cover Graph William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 2/18

3 Diagrams and cover graphs f e e b f e c d c d f c d a b a b a Three different posets with the same cover graph. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 3/18

4 Incidence poset e ab bd cd ac ae de d b c G a a b c d e PG The incidence poset P G of G has V E as its ground set; vertices in V are minimal elements in P G, edges in E are maximal elements in P G. Vertex x is covered by edge e in P G if and only if x is one of the endpoints of e in G. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 4/18

5 Dimension of a poset The dimension dim(p) of a poset P is the least positive integer for which there are t linear orders L 1,L 2,...,L t on the ground set of P so that x y in P if and only if x y in L i for each i = 1,2,...,t. c f d g e L 1 = b < e < a < d < g < c < f L 2 = a < c < b < d < g < e < f L 3 = a < c < b < e < f < d < g a b William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 5/18

6 Dimension of the incidence poset of G The dimension dim(p G ) of the incidence poset of G, can alternatively be defined as the least positive integer t for which there is a family {L 1,...,L t } of linear orders on V such that the following two conditions are satisfied: If x, y and z are distinct vertices of G and {y,z} is an edge in G, then there is some i with x > y and x > z in L i. If x and y are distinct vertices of G, then there is some i with x > y in L i. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 6/18

7 Schnyder s Theorem Theorem (Schnyder, 1989) A graph G is planar if and only if dim(p G ) 3. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 7/18

8 Schnyder s Theorem Theorem (Schnyder, 1989) A graph G is planar if and only if dim(p G ) 3. Schnyder s Theorem characterizes planar graphs in terms of the dimension of their incidence poset. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 7/18

9 Questions Let P G be incidence poset of graph G. Haxell asked the following question: Question Is χ(g) bounded in terms of dim(p G )? William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 8/18

10 Questions Let P G be incidence poset of graph G. Haxell asked the following question: Question Is χ(g) bounded in terms of dim(p G )? Dual Question Is dim(p G ) bounded in terms of χ(g)? William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 8/18

11 Yes to the dual question Theorem (Agnarsson, Felsner and Trotter, 1999) Let G be a graph and let P G be the incidence poset of G, then dim(p G ) = O(lglgr) where χ(g) = r. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 9/18

12 Yes to the dual question Theorem (Agnarsson, Felsner and Trotter, 1999) Let G be a graph and let P G be the incidence poset of G, then dim(p G ) = O(lglgr) where χ(g) = r. The bound here is best possible, and it can be argued using the classic theorem of Erdös and Szekeres. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 9/18

13 No to the original question For original question, if dim(p G ) 3, then by Schnyder s theorem, G is planar. Hence χ(g) 4. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 10/18

14 No to the original question For original question, if dim(p G ) 3, then by Schnyder s theorem, G is planar. Hence χ(g) 4. However, in general, we have the following theorem: Theorem (Trotter and Wang / Mendez and Rosenstiehl, 2005) For every r 1, there exists a graph G with χ(g) r and dim(p G ) 4. The inequality dim(p G ) 4 is tight once r = 5. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 10/18

15 A theorem of Kříž and Nešetřil Theorem (Kříž and Nešetřil, 1991) For every r 1, there exists a poset P so that dim(p) = 2 and χ(g P ) r. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 11/18

16 eye(g) Definition Given a graph G, eye(g) is the least positive integer t for which there exists a family {L 1,...,L t } of linear orders on the vertex set of G such that if x, y and z are distinct vertices of G and {y,z} is an edge in G, then there is some i for which x is not between y and z in L i. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 12/18

17 eye(g) Definition Given a graph G, eye(g) is the least positive integer t for which there exists a family {L 1,...,L t } of linear orders on the vertex set of G such that if x, y and z are distinct vertices of G and {y,z} is an edge in G, then there is some i for which x is not between y and z in L i. Corollary eye(g) dim(p G ) 2eye(G). William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 12/18

18 Classic results Theorem (Kříž and Nešetřil, 1991) For every r 1, there exists a graph G so that eye(g) 2 and χ(g) r. Theorem (Kříž and Nešetřil, 1991) For every pair (g,r), there is a graph G with girth(g) g, χ(g) r and eye(g) 3. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 13/18

19 A question Kříž and Nešetřil s question For every pair (g,r), does there exist a graph G with girth(g) g, χ(g) r and eye(g) 2? William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 14/18

20 A question Kříž and Nešetřil s question For every pair (g,r), does there exist a graph G with girth(g) g, χ(g) r and eye(g) 2? Erdös-Hajnal conjecture, 1989 Every graph with sufficiently large chromatic number contains a subgraph which has large girth and large chromatic number. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 14/18

21 A question Kříž and Nešetřil s question For every pair (g,r), does there exist a graph G with girth(g) g, χ(g) r and eye(g) 2? Erdös-Hajnal conjecture, 1989 Every graph with sufficiently large chromatic number contains a subgraph which has large girth and large chromatic number. If the answer to the question of Kříž and Nešetřil is No, then the Erdös-Hajnal conjecture fails. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 14/18

22 Our answer Theorem (Trotter and Wang) For every g,r 1, there exists a poset P = P(g,r) with cover graph G = G(g,r) so that the height of P is r, while girth(g) g and χ(g) = r. Furthermore, there are two linear extensions L 1 and L 2 of P witnessing that eye(g) 2. Our proof uses the construction of Nešetřil and Rödl for hypergraphs with large girth and large chromatic number. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 15/18

23 Open questions Question Is it true that for every pair (g,r) of integers, there is a poset P so that dim(p) 2, the girth of the cover graph G of P is at least g and the chromatic number of G is at least r? William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 16/18

24 Open questions Question Is it true that for every pair (g,r) of integers, there is a poset P so that dim(p) 2, the girth of the cover graph G of P is at least g and the chromatic number of G is at least r? We believe the answer is no. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 16/18

25 Conjectures Conjecture For every pair (g,d) of integers, with g 5 and d 1, there is an integer r = r(g,d) so that if G is the cover graph of a poset P, dim(p) d and girth(g) g, then χ(g) r. William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 17/18

26 Thank you! William T. Trotter, Ruidong Wang, Incidence Posets and Cover Graphs, 18/18