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### Acyclic Orientations

### The End

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

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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.

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.

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).

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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

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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.

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.

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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

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.

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.

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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 graphsA 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).

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).

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 xV(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.)

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}.

Circular Coloring

- Suppose positive integers m and k with m2k. An (m,k)-coloringf of a graph G=(V,E) is a function from V to {0,1,…,m-1} such that if uvE then

k |f(u)-f(v)| m-k.

- The circular chromatic number

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

The Set of Acyclic Orientations

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

r(C)=max{m/k, m/(mk)}.

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

Circular Chromatic Numbers

- We proved that c(G)= (G).
- That is, there exists DOa(G) such that (D) =(G) and there exists a cycle C in D such that

r(C)=(D)= (G)=c(G).

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 xyE(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’.

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=SiV(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 mnk/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).

Chip Firing and Circular Coloring

- r(C)=(D)c(G) by definition.
- Therefore c(G)=pe(G).

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.

Thank You

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