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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 on Graph Theory January 10-14, 2009 Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan http://mail.math.nsysu.edu.tw/~comb/2009/

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Study from chip firing game to cover graph l.jpg

Study from chip-firing game to cover graph

Li-Da Tong

National Sun Yat-sen University

August 12, 2008


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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.


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Outline

  • Chip-firing games

  • Acyclic orientations

  • Cover graphs

  • Fully orientable graphs

  • The relation between chip firing and circular coloring


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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)


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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.

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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.


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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.


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Acyclic Orientations

Chip-Firing Games with |E(G)| Chips


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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.


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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


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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).

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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

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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.


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Hasse diagram

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



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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.

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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



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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.



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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.


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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.


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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.


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Graph DFS treeOrientation

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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.


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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.

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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


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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.

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Non-fully-orientable graphs

  • Kr(n) : the complete r-partite graph such that each part has n vertices.

    K3(2)


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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


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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.


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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?


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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.


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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.


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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.



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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.


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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.


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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).


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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).


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Chip-Firing with |E(G)| Chips

Theorem: If the number of chips of a periodic configuration c is |E(G)| then there exists a unique acyclic orientation D of G such that the outdegree of x is c(x) for all xV(G).

(Sketch of proof: Since c is periodic, there is a permutation (v1,v2,…,vn) of all vertices of G such that c is invariant under firing vertices by the order v1,v2,…,vn. Then we have an acyclic orientation D with vi being a source of Di where D1=D and Di=Di-1{v1,v2,…,vi-1} for i=2,3,…,n.)


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Chip-Firing with |E(G)| Chips

  • If S is a set of sources of D then Fs(D) is an acyclic orientation obtained from D by reversing all arcs from vertices of S.

  • (S1,S2,…,Sm) is called a (m,k)-sequence of D if FS1,S2,…,Sm(D)=D.

  • Let D be am acyclic orientation of G. Then we define p(D)=min{m/k: there exists a (m,k)-sequence of D}.


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Circular Coloring

  • Suppose positive integers m and k with m2k. An (m,k)-coloringf of a graph G=(V,E) is a function from V to {0,1,…,m-1} such that if uvE then

    k |f(u)-f(v)| m-k.

  • The circular chromatic number

    c(G)=inf{m/k: G has an (m,k)-coloring}.


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The Set of Acyclic Orientations

  • Let Oa(G) be the set of all acyclic orientations of G and DOa(G). If C is a cycle of D with k clockwise arcs and mk anti-clockwise arcs the we define

    r(C)=max{m/k, m/(mk)}.

  • (D)=max{r(C): C is a cycle of D} and (G)=min{(D): D Oa(G)}.


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Circular Chromatic Numbers

  • We proved that c(G)= (G).

  • That is, there exists DOa(G) such that (D) =(G) and there exists a cycle C in D such that

    r(C)=(D)= (G)=c(G).


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Chip Firing and Circular Coloring

  • c(G)pe(G).

    Since c(G)=m/k, there is an (m,k)-coloring f.

    Define an acyclic orientation D’ of G by

    (x,y) if and only if xyE(G) and f(x)<f(y).

    Let Vi= f -1(i) for i=0,1,2,…,m-1.Then V0, V1, …, Vm-1 are independent sets. By Vj∪Vj+1∪ … ∪Vj+k-1: an independent set for all j, we have an (m,k)-sequence for D’.


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Chip Firing and Circular Coloring

  • pe(G)=p(D) for some D Oa(G).

    Let C be a cycle of D with maximum r(C).

  • There exists an (m,k) sequence s=(S1,S2,…,Sm) forD.

  • Let Ti=SiV(C) for i=1,2,…,m. Then (T1,T2,…,Tm) is an(m,k) sequence for C. Then p(D)p(C).

  • Suppose C has n vertices and t clockwise arcs.

  • Then |Ti|min{t,n-t} and mnk/max|Ti|.

  • So m/k n/max|Ti|.

  • By max|Ti|min{t,n-t}, m/k  n/min{t,n-t}=r(C).

  • Hence p(C)r(C).


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Chip Firing and Circular Coloring

  • r(C)=(D)c(G) by definition.

  • Therefore c(G)=pe(G).


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Chip-firing and Coloring

  • Goddyn, Luis A.; Tarsi, Michael; Zhang, Cun-QuanOn $(k,d)$-colorings and fractional nowhere-zero flows.J. Graph Theory28 (1998), no. 3, 155--161.

  • Yeh, Hong-Gwa; Zhu, XudingResource-sharing system scheduling and circular chromatic number.Theoret. Comput. Sci.332 (2005), no. 1-3, 447--460.


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The End

Thank You


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