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1. Study from chip-firing game to cover graph Li-Da Tong National Sun Yat-sen University August 12, 2008

2. 2009 Workshop onGraph TheoryJanuary 10-14, 2009Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiung, Taiwan • http://mail.math.nsysu.edu.tw/~comb/2009/ • First AnnouncementSponsored by National Sun Yat-sen University, Institute of Mathematics of Academia Sinica, National Center for Theoretical Sciences(South), 2009 Workshop on Graph Theory will be held in the Department of Applied Mathematics, National Sun Yat-sen University in Kaohsiung, Taiwan. Discrete Mathematics is an active research area in Taiwan. The aim of the workshop is to provide a platform for the participants to exchange ideas, results and problems. The workshop is expected to attract about 30 participants from abroad and 120 participants from Taiwan.

3. Outline • Chip-firing games • Acyclic orientations • Cover graphs • Fully orientable graphs • The relation between chip firing and circular coloring

4. Chip Firing Games Chip-firing games were first introduced by Björner et al.(Björner, Anders, Lovász, László and Shor, Peter W. Chip-firing games on graphs. European J. Combin.12 (1991), no. 4, 283—291)

5. Chip Firing Games A chip-firing game is played on a graph G with a nonnegative integer function c from V(G) to Z. Let vV(G). Then c is called a configuration of G and c(v) is the number of chips on the vertex v. A fire on v is the process that each neighbor of v gets one chip from v. A fire on a 2 0 a a 1 0 2 1 b e b e 2 2 c d c d 0 0

6. Chip Firing Games In the game, we restrict that a vertex v can be fired on a function c if and only if deg(v)c(v). The game continues as long as fires exist. If the number of chips is greater than 2|E(G)||V(G)|, then the game is infinite. If the number of chips is less than |E(G)|, then the game is finite. For the number of chips between |E(G)| and2|E(G)||V(G)|, the length of a game is determined by the initial distribution of chips.

7. Chip Firing Games • Björner, Anders; Lovász, László; Shor, Peter W.Chip-firing games on graphs.European J. Combin.12 (1991), no. 4, 283--291. • Eriksson, KimmoNo polynomial bound for the chip firing game on directed graphs.Proc. Amer. Math. Soc.112 (1991), no. 4, 1203--1205. • Tardos, GáborPolynomial bound for a chip firing game on graphs.SIAM J. Discrete Math.1 (1988), no. 3, 397--398. • Bitar, Javier; Goles, EricParallel chip firing games on graphs.Theoret. Comput. Sci.92 (1992), no. 2, 291--300. • Björner, Anders; Lovász, LászlóChip-firing games on directed graphs.J. Algebraic Combin.1 (1992), no. 4, 305--328.

8. Acyclic Orientations Chip-Firing Games with |E(G)| Chips

9. The Number of Chips is |E(G)| • Let c be a configuration of G and the number of chips be vV(G)c(v)=|E(G)|. • Let v1,v2,…,vk be vertices of G. Then Fv1,v2,…,vk(c)=d if d is obtained from c by firing vertices in the ordering v1,v2,…,vk. • c is called periodic if there exists a permutation s on V(G) such that Fs(c)=c.

10. The Number of Chips is |E(G)| If a game with an initial configuration c and |E(G)| chips is infinite, then there exist a sequence s of vertices and a periodic configuration p such that Fs(c)=p. Firing sequence x1,x2,…,xk, v1,v2,…,vn Configuration c p p

11. Periodic configurations • For every periodic configuration c, there exists a permutation s:v1,v2,…,vn of V(G) such that Fs(c)=c. • Then there exists an acyclic orientation D of G such that the out-degree of v in D=c(v) for vV(G). 1 1 2 0 0 1 1 2 1 b b a a b b a a 1 1 c c 0 0 c 1 c 0 0

12. Acyclic orientations By the permutation s:v1,v2,…,vn of V(G), vivjE(G) and i<j if and only if (vi,vj) A(D). Firing at a vertex Reversing a source 1 2 b a b a c 0 c

13. 2 1 b a b a c 0 c 0 2 b a b a c 1 c 1 0 b a b a c 2 c

14. Cover Graphs

15. Cover Graphs • A cover graph is the underlying graph of the Hasse diagram of a finite partially ordered set. • Given an acyclic orientation of a graph, Edelmen defined an arc to be dependent if its reversal creates a directed cycle. • A graph is a cover graph if and only there exists its acyclic orientation without dependent arcs.

16. Hasse diagram The power set of {x, y, z}, partially ordered by inclusion.

17. The underlying graph of the previous Hasse diagram.

18. Cover Graphs • If D is an acyclic orientation of G without dependent arcs and D’ is obtained from D by reversing a source, then D’ is also an acyclic orientation of G without dependent arcs. a a b e b e c d c d

19. Cover Graphs Theorem. If G is a cover graph, there exists an acyclic orientation without dependent arcs having a uniquely fixed source. Grötzsch’s graph

20. We find a second source! It is a contradiction!

21. Cover Graphs (Brightwell, 1993) The recognition problem of cover graphs is NP-complete. (Nešetřil, Rödl, 1978) There are non-cover graphs with arbitrarily large girth. Concrete examples of non-cover graphs have been constructed for only graphs having girth at most 6.

22. Fully orientable graphs

23. d(D), dmin(G), dmax(G) d(D) : the number of dependent arcs of an acyclic orientation D. dmin(G) : the minimum number of dependent arcs over all acyclic orientations of G. dmax(G) : the maximum number of dependent arcs over all acyclic orientations of G.

24. dmax(G) Theorem. If G is a graph with k components, then dmax(G) = ||G||－ |G| ＋ k. ||G|| : number of edges |G| : number of vertices k : number of components D. C. Fisher, K. Fraughnaugh, L. Langley, and D. B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory, Ser. B, 71(1997), 73–78.

25. Step 1 An acyclic orientation with ||G||  |G| + 1 dependent arcs can be constructed by orienting edges away from the root of a depth-first search tree. (Every non-tree edge joins a vertex with one of its ancestors.) Step 2 Every acyclic orientation of G contains a spanning tree of G when all dependent arcs are removed. dmax(G) = ||G||－ |G| ＋ 1.

26. Graph DFS treeOrientation 1 1 2 2 3 6 3 6 7 7 4 5 4 5 dmax(G) = ||G||－ |G| ＋ 1.

27. Fully orientable graphs A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G.

28. Fully orientable graphs A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4

29. Fully orientable graphs A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4

30. Fully orientable graphs A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4

31. Non-fully-orientable graphs • Kr(n) : the complete r-partite graph such that each part has n vertices. K3(2)

32. Non-fully-orientable graphs Theorem. (Chang, Lin, Tong)For r 3 and n2, the complete r-partite graphs Kr(n) are not fully orientable.K3(2) 4, 5, 6, 7

33. Fully orientability for Dmin(G)  1 Theorem. (Lai, Lih, Tong) If G is a connected graph with dmin(G)  1, then G is fully orientable.

34. Open Problems Question 1 For any given integer g 4, does there exist a non-fully-orientable graph G whose girth is equal to g? Question 2 Does there exist a non-fully-orientable graph G whose dmin(G) is equal to 2 or 3? Question 3 K3(2) shows that a maximal planar graph can be non-fully-orientable. Howto characterize all fully orientable planar graphs?

35. Related Papers • West, Douglas B.Acyclic orientations of complete bipartite graphs.Discrete Math.138 (1995), no. 1-3, 393--396. • Fisher, David C.; Fraughnaugh, Kathryn; Langley, Larry; West, Douglas B.The number of dependent arcs in an acyclic orientation.J. Combin. Theory Ser. B71 (1997), no. 1, 73--78. • Rödl, V.; Thoma, L.On cover graphs and dependent arcs in acyclic orientations.Combin. Probab. Comput.14 (2005), no. 4, 585--617. • K.-W. Lih, C.-Y. Lin, and L.-D. Tong, On an interpolation property of outerplanar graph, Discrete Applied Mathematics154 (2006) 166-172. • K. W. Lih, C.-Y. Lin, and L.-D. Tong, Non-cover Generalized Mycielskian, Kneser, and Schrijver graphs, Discrete Mathematics (2007), doi:10.1016/j.disc. 2007.08.082. • G. J. Chang, C.-Y. Lin, and L.-D. Tong, The independent arcs of acyclic orientations of complete r-partite graphs, revised.

36. Related Papers • H.-H. Lai, G. J. Chang, K.-W. Lih, On fully orientability of 2-degenerate graphs, Inform. Process. Lett. 105(2008), 177-181. • H.-H. Lai, K.-W. Lih, On preserving fully orientability of graphs, European J. Combin., to appear. • H.-H. Lai, K.-W. Lih, The minimum number of dependent arcs and a related parameter of generalized Mycielski graphs, manuscript. • H.-H. Lai, K.-W. Lih, C.-Y. Lin, L.-D. Tong, When is the direct product of generalized Mycielski graphs a cover graph? manuscript. • H.-H. Lai, K.-W. Lih, L.-D. Tong, Fully orientability of graphs with at most one dependent arc, manuscript. • O. Pretzel, On graphs that can be oriented as diagrams of ordered sets, Order 2(1985), 25-40. • O. Pretzel, On reorienting graphs by pushing down maximal vertices, Order 3(1986), 135-153.

37. Related Papers • O. Pretzel, On reorienting graphs by pushing down maximal vertices II, Discrete Math. 270(2003), 227-240. • K. L. Collins, K. Tysdal, Dependent edges in Mycielski graphs and 4-colorings of 4-skeletons, J. Graph Theory 46(2004), 285-296. • P. Holub, A remark on covering graphs, Order 2(1985), 321-322.

38. The Relation between Chip Firing and Circular Coloring

39. Chip-firing on an independent set • Let S be an independent set in G. The chip firing on S is the process as sending one chip to every neighbor of each vertex in S. • A configuration c is called periodic if there exist independent sets S1, S2,…,Sm such that FSm(FSm-1(…FS1(c)…))=c. To simplify the notation FSm(FSm-1(…FS1(c)…))= FS1S2…Sm(c). Such sequence (S1,S2,…,Sm) is called a periodsequence of c.

40. Chip Firing Games • Lemma: If c is a periodic configuration for a connected graph G and FS1S2…Sm(c)=c then every vertex of G occurs in the same number of sets in {S1,S2,…,Sm}. Proof. Let n(v) be the number of sets containing the vertex v in {S1,S2,…,Sm}. Take v in G with n(v): maximum. Let NG(v)={u1,u2,…,ur}. By FS1S2…Sm(c)=c, rn(v)= n(u1)+n(u2)+…+n(ur). By n(v): maximum, n(v)= n(u1)=n(u2)=…=n(ur). By G: connected, n(u) is a comstant.

41. Chip Firing Games • A periodic sequence is called a (m,k)-sequence if its length is m and each vertex occurring (or fired) in exactly k sets (times).

42. 0 2 a a 2 2 1 0 b e firing b e on {a,d} c d 0 2 1 c d 0 firing b,e c,e 2 1 0 a a a 2 1 0 0 0 2 b e b e b e a,c b,d c d 1 2 2 c d 0 c d 2 0

43. The Number of Chips is |E(G)| • We restrict that the number of chips of each periodic configuration is |E(G)|. Define that pe(G)=inf{n/k: G has an (n,k)-configuration with |E(G)|chips}. • In the following, we will show that pe(G)= c(G).