Study from chip-firing game to cover graph

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## Study from chip-firing game to cover graph

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**Study from chip-firing game to cover graph**Li-Da Tong National Sun Yat-sen University August 12, 2008**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.**Outline**• Chip-firing games • Acyclic orientations • Cover graphs • Fully orientable graphs • The relation between chip firing and circular coloring**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)**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 vV(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**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.**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.**Acyclic Orientations**Chip-Firing Games with |E(G)| Chips**The Number of Chips is |E(G)|**• Let c be a configuration of G and the number of chips be vV(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.**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**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 vV(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**Acyclic orientations**By the permutation s:v1,v2,…,vn of V(G), vivjE(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**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**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.**Hasse diagram**The power set of {x, y, z}, partially ordered by inclusion.**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**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**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.**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.**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.**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.**Graph DFS treeOrientation**1 1 2 2 3 6 3 6 7 7 4 5 4 5 dmax(G) = ||G||－ |G| ＋ 1.**Fully orientable graphs**A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G.**Fully orientable graphs**A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4**Fully orientable graphs**A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4**Fully orientable graphs**A graph is fully orientable if every number d satisfying dmin(G) ddmax(G) is achievable as d(D) for some acyclic orientation D of G. 2, 3, 4**Non-fully-orientable graphs**• Kr(n) : the complete r-partite graph such that each part has n vertices. K3(2)**Non-fully-orientable graphs**Theorem. (Chang, Lin, Tong)For r 3 and n2, the complete r-partite graphs Kr(n) are not fully orientable.K3(2) 4, 5, 6, 7**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.**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?**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.**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.**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.**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.**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.**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).**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**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).