The nowhere-zero 6-flow theorem of Seymour is proven by construction

Hoede, C.

Uttunggadewa, S.

English

Uttuggadewa, S.

NARCIS 

An alternative proof of the nowhere-zero 6-flow theorem

University of Twente Research Information

Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1564An alternative proof of thenowhere-zero 6-flow theoremC. Hoede and S. UttunggadewaJanuary 2001ISSN 0169-2690An Alternative Proof of the Nowhere-zero6-flow TheoremC. Hoede and S. UttunggadewaFaculty of Applied MathematicsUniversity of TwenteP.O. Box 2177500 AE Enschede, The NetherlandsAbstractThe nowhere-zero 6-flow theorem of Seymour is proven by construction.Key words: bridgeless graph, nowhere-zero 6-flow, construction.AMS Subject Classification: 05C38, 05C99.1 IntroductionWe consider connected cubic bridgeless graphs. For terminology we refer to thebook of Holton and Sheehan [1].Definition 1.1 A graph G is a cycle dominated graph if G has a cycleC such thatG \ V C is acyclic.Notice that if a graphG has a dominating cycle, thenG is a cycle dominatedgraph.Lemma 1.1 Every cubic hamiltonian graph has a nowhere-zero 6-flow with oddflow value on each edge of its hamiltonian cycle.Proof The number of vertices of a cubic hamiltonian graph is even. We cancolour the edges in its hamiltonian cycle by two coloursx andy alternatingly.1Colour the remaining edges by the colourz. Send a flow of value 1 throughxz-cycles and a flow of value 3 throughyz-cycles in arbitrary direction. Since eachedge in the cycle is covered once, the flow value on each edge of the cycle is 1 or3. Lemma 1.2 If a cubic bridgeless graph G has a cycle C such thatG \ V C isacyclic, then G has a nowhere-zero 6-flow.Proof Let us consider a cycle dominated graphG. There is a cycleC such thatG \ V C is a forest. Since a forest is 2-vertex colourable, we can orient the edgesof G such that each vertex inV G \ V C becomes a sink or source, see Figure 1.rr rrr @? I@  I@@?I@ I @	R rr rrrr @? I@@  @I @@?I@ I @	R@ R 	Figure 1: Sinks and sources in a forest.Send a flow of value 2 through each edge that is incident with a vertex inV G \ V C according to the orientation. Total inflow and total outflow at eachvertex are equal to 0 modulo 6.We can find a 6-flowf1 onG such that each edge incident with a vertex inV G \ V C has a flow of value 2, each edge ofC has a flow of value 0, 2 or 4, andeach remaining edge has a flow of value 0. This flowf1 is constructed by dealingwith the forestG\V C in the way described. The vertices ofC adjacent to a vertexof V G \ V C divideC into parts. Choose one of these parts and send a flow ofvalue 0 in an arbitrary direction, i.e. simply choose flow value 0 on this part. Theflow values on the other parts are now determined. This construction is consistentsince the sum of the flows at vertices ofG \ V C is equal to 0 modulo 6 and so isthe sum of inflow and outflow values from and to these vertices for vertices ofC.We want all orientations to be chosen towards the cycleC for edges incident to2vertices ofC. This means that we have to change flow values from 2 to 4 whenthe orientation is to be changed. Call these flow valuesai, i = 1, 2, ..., p. a1 andap are inflow values for the vertices ofC determining the part that is chosen toget a flow of value 0. By the chosen orientations the flow values must be∑ki=1 ai,k = 1, 2, ..., p on the edges of the cycleC. In particular the flow on the partbetweenap anda1 must be∑pi=1 ai, but we argued that this is equal to 0 modulo6.ConsiderG < V C >, the subgraph ofG induced byV C. SinceC is a cycle,it is clear thatG < V C > is a hamiltonian graph. Note thatG < V C > is notcubic anymore. LetC∗ be the hamiltonian graph derived fromG < V C > bycontracting edges incident with vertices of degree 2. By Lemma 1.1, we can finda nowhere-zero 6-flowf ∗2 with odd flow value on each edge of its cycle. For anexample, see Figure 2.SinceG < V C > is a subdivision ofC∗,G < V C > also has a nowhere-zero6-flow f2 with odd flow value on each edge of its cycle. SinceG < V C > is asubgraph ofG, f2 can be used to construct a 6-flow onG by putting a flow of value0 on each edge that is incident with a vertex inV G \ V C . It is clear thatf1 + f2is a nowhere-zero 6-flow on the graphG. Notice that each edge inEG \ EC hasa flow of value 2 or 4 and each edge inEC has a flow of value 1, 3 or 5. ss ss ss ssHHHHAAAAAHHHHAAAAALLLLLLLLLLLLLLLLLLLLLL         HY1AK31*3Y1AK313K2U 4-2 	2Figure 2: A nowhere-zero 6-flowf ∗2 on a hamiltonian graphC∗ derived fromG〈V C〉 by contracting edges incident with vertices of degree 2.32 Nowhere-zero 6-flowsLet G be a bridgeless graph. A familyC of cycles{C1, C2, ... , Cs} of G isa spanning cycle of G if | V G | = | V C |≡|⋃si=1 V Ci |. Note thatEC =⋃si=1ECi.A cubic graphG is a cycle dominated graph ifG has a spanning cycle con-sisting of a single cycle. It has been proven in Lemma 1.2 that a cycle dominatedgraph has a nowhere-zero 6-flow. The important point in the proof of this lemmais that we can colour the verticesG \ V C by two colours and make them sinksand sources.Definition 2.1 A cubic graph G is called an almost cycle dominated graph if Ghas a spanning cycle{Cm, C1, C2, ... ,Cs} such that if we contractC1, C2, ... ,Cs and removeCm, the rest is a forest.Lemma 2.1 An almost cycle dominated graph has a nowhere-zero 6-flow.Proof SupposeC = {Cm, C1,C2, ... ,Cs} is a spanning cycle ofG such that if wecontractC1, C2, ... ,Cs and removeCm, the rest is a forest. Remove all chords ofthe cycles ofC. If we removeCm and one edge of each other cycle ofC, then therest is a forest. By the construction of a 6-flowf1 as in the proof of Lemma 1.2,we can find a 6-flowf11 of G such that each chord of a cycle ofC has a flow ofvalue 0, each edge in a cycle ofC has a flow of value 0, 2 or 4 modulo 6 and eachremaining edge has a flow of value 2.Again, by the construction of a 6-flowf2 as in the proof of Lemma 1.2, wecan find a 6-flowf22 of G such that each chord of a cycle ofC has a flow of value2 and an odd flow value on each edge of a cycle ofC. It is clear thatf11 + f22 is anowhere-zero 6-flow on the graphG. Notice that each edge of a cycle ofC has aflow of value 1, 3 or 5 and the other edges have a flow of value 2. LetA be a configuration, as in Figure 3.a, that is contained in a cubic graph.Define a configurationB by removing the edges (v1,v2) and (v3,v4) and addingedges (v1,v4) and (v2,v3), See Figure 3.b. This transformation we call ashiftfromA toB.4&%'$&%'$ss@    @@@  @@sssse1e2e3Configuration B(3.a)ss@   @@sssse1e2e3Configuration A(3.b)v1 v1v3 v3v2 v2v4 v4=⇒Figure 3: A shift.Lemma 2.2 Consider a graphG that contains configurationA. Define a graphG∗ by a shift fromA toB. If G has a nowhere-zero 6-flow such that(v1, v2) and(v3, v4) have a flow of value 2 ande2 has an odd flow value in arbitrary direction,thenG∗ has a nowhere-zero 6-flow such that(v1, v4) and (v3, v2) have a flow ofvalue 2 in the same direction as(v1, v2) and(v3, v4) ande1 still has an odd flowvalue.Proof If in G (v1, v2) and(v3, v4) have a flow of value 2 in the same direction,then nothing is changed. If inG the flows of value 2 in(v1, v2) and(v3, v4) arenot in the same direction, then to the flow value ofe1 4 is added fromv2 to v4.&%'$&%'$ss@    @@@@I  @@sssse1e2e3ss@ -  @@sssse1e2e3v1 v1v3 v3v2 v2v4 v4=⇒Figure 4: Flow of value 2 in(v1, v2) and(v3, v4) not in the same direction.A consequence of Lemma 2.2 is that, for our nowhere-zero 6-flow problem,the position of vertices on a cycle ofC is not important.5Our main goal is to prove Theorem 2.1, the 6-flow theorem, by construction.We will give an outline of the proof by dealing with an example.Consider a cubic graph G and its spanning cycleC = {Cm, C1, C2, C3, C4}.See Figure 5.'&$%.......................................................................... .......................................................................... ...................................................................................................................................................ss ssss s s s s s s s s s ssss ss s sss ss ssss sCmC1 C2 C3C4%%%%%        AAAA............................................................................................................................................................................................................................................................""""""".....................................................................................................      Figure 5: General structure of a cubic graph G and its spanning cycleC = {Cm,C1, C2, C3, C4}.G is not an almost cycle dominated graph since after we contractC1, C2, C3,C4 still a cycle exists even when removingCm , see Figure 6. Cycles(C1, C2, C4, C1)and(C2, C3, C2) are created.'&$%x xxxs s s s s s s s s s s     AAAAABBBBBBBB.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................CmC1C2C3C4Figure 6: A typical structure after contracting cycles.6If in Gwe remove arbitrary edges that correspond to an edge of cycle(C1, C2, C4, C1)and an edge of cycle(C2, C3, C2), we get, after contracting edges of degree 2, analmost cycle dominated graphG∗. See Figure 7. This procedure to get an almostcycle dominated graph can be applied in general.'&$%......................................................................... ......................................................................... ..................................................................................................................................................ss ssss s s s s s s s s s ssss s ssss ssssCmC1 C2 C3C4%%%%%        AAAA"""""".............................................................................................................................................................................................................................................................."""""""      Figure 7: An almost cycle dominated graphG∗.By construction of a nowhere-zero 6-flow as in Lemma 2.1, a nowhere-zero6-flow f1 of G∗ can be constructed the directions of which can be seen in Figure8. An arrow on an edge indicates a flow of value 2. The other edges, ofCm, C1,C2, C3 orC4, have an odd flow value.'&$%......................................................................... ......................................................................... ..................................................................................................................................................sss sss s s s s s s s s s ssss s ssss ssssCmC1 C2 C3C4%%%%%        AAAAU ?""""""........................................................................?.....................................................................................................................................................................................I"""""""3-      :Figure 8: A nowhere-zero 6-flowf1 on an almost cycle dominated graphG∗.7Consider Figure 9. To get a nowhere-zero 6-flow ofG, we have to send aflow of value 2 around a cycle that contains edge(v5, v6) and around a cycle thatcontains(v8, v9). Note that by sending a flow of value 2, the flow value of eachedge in the cyclesCm, C1, C2, C3 orC4 is still odd.'&$%......................................................................... ......................................................................... ...................................................................................................................................................ss ssss s s s s s s s s s ssss ss s sss ss ssss sCmC1 C2 C3C4%%%%%        AAAA""""""..........................................................................................................................................................................................................................................................................."""""""............................................................................................................       U 6  I3-06:0v1v2v3v4v5v6v7v8v9v10v11v12 v13Figure 9: A 6-flow ofG.By sending a flow of value 2 around cycle(v7, v11, v10, v9, v8, v6, v7), a flowof value 2 on edge(v7, v11) is changing to 4 or is still 2, in the opposite direction,and the flow of value 0 on edge(v9, v8) is changing to 2.We cannot send a flow of value 2 around cycle(v6, v5, v4, v3, v2, v1, v13, v8, v6)since a flow of value 0 on(v2, v3) is created.We can solve this problem by shifting the positions ofv2 andv12 before weuse the construction method of Lemma 2.1. After we get a nowhere-zero 6-flowfor the resulting graph, we can shift the positions back. The result,f2, can be seenin Figure 10. Note that various flow directions have changed.8'&$%......................................................................... .......................................................................... ..................................................................................................................................................sss sss s s s s s s s s s ssss s ssss ssssCmC1 C2 C3C4%%%%%	        	AAAAU ?""""""........................................................................?.....................................................................................................................................................................................R"""""""+-      :Figure 10: A nowhere-zero 6-flowf2 on an almost cycle dominated graphG∗.We finally consider Figure 11. Note that, in particular, the direction of theflow on (v3, v2) has changed. From the nowhere-zero 6-flow of the almost cycledominated graphG∗, we can now find a nowhere-zero 6-flow ofG by sendinga flow of value 2 around cycle(v6, v5, v4, v3, v2, v1, v13, v8, v6) and also a flow ofvalue 2 around cycle(v7, v11, v10, v9, v8, v6, v7), to obtain a nowhere-zero 6-flowfor the original graph.'&$%......................................................................... ......................................................................... ...................................................................................................................................................ss ssss s s s s s s s s s ssss ss s sss ss ssss sCmC1 C2 C3C4%%%%%        AAAA""""""............................................................................................................................................................................................................................................................................."""""""............................................................................................................      		U 6  R+-06:0v1v2v3v4v5v6v7v8v9v10v11v12v13Figure 11: A 6-flow onG.9Theorem 2.1 (Seymour [2])A bridgeless cubic graph has a nowhere-zero 6-flow.Proof The construction steps carried out in the example can be carried out in thegeneral case as well.There are seven steps in the construction of the nowhere-zero 6-flow :1. Finding a spanning cycleC = {C1, C2, C3, ..., Cn}.2. Removing all chords of each cycle ofC.3. ChoosingCm, a cycle ofC, and contracting the other cycles.4. RemovingCm. If there is a cycleCk, remove an edge of this cycle. Notethat the rest is a forest.5. Applying the construction of a 6-flow of Lemma 2.1. By the shifting method,the direction of flow values on the edges ofCk can be forced to be in thesame direction.6. Sending a flow of value 2 around theCk in the same direction of the direc-tion of flow value 2 of the edges ofCk.7. Applying the construction of Lemma 1.1, so that all chords have a flow ofvalue 2.References[1] J. A. Holton and J. Sheehan.The Petersen Graph. Cambridge UniversityPress, Cambridge, 1993.[2] P. Seymour. Sums of circuits. In J. Bondy and U. S. R. Murty, editors,Graph Theory and Related Topics, pages 431–355. Academic Press, NewYork, 1979.10

Sums of circuits.

The Petersen Graph.

https://ris.utwente.nl/ws/files/5113225/1564.pdf

An alternative proof of the nowhere-zero 6-flow theorem

Abstract

Similar works

Full text

Available Versions

NARCIS

University of Twente Research Information