graphs with 1-factors

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University of South Carolina

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1-1-1974

Graphs with 1-FactorsDavid SumnerUniversity of South Carolina - Columbia, [email protected]

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Publication InfoProceedings of the American Mathematical Society, Volume 42, Issue 1, 1974, pages 8-12.

1974 by American Mathematical Society
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PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 42, Number 1, January1974

GRAPHS WITH 1-FACTORSDAVID P. SUMNER

ABSTRACT.n this paper it is shown that if G is a connectedgraph of order 2n (n> 1) not containinga 1-factor, hen for each k,1


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GRAPHS WITH 1-FACTORS 9that G* =G- {x, y} is not a connectedgraph.Let A be the componentofG* that containsa. We note that since D is a diameter f G, everypoint inG*-A is adjacento x. Thus f G* A containsa nontrivial omponentB,then for any b, c EB whichare adjacent, verypoint n G -{b, c} is joinedto x by a path, and hence G-{b, c} is connected.Similarly,f thereexistsa point e E G*-A whichis adjacent o y, then G-{e, y} is connected.Hencewemay suppose hat G* A containsonly isolatedpointsand eachof these points is adjacentonly to x. Thusevery point in G*-A is anendpointof Gwith ointx. Therefore inceGhasno coincident ndpoints,G*-A mustconsistof a single isolatedpointf Ify werean endpointofG, thenf andy wouldbe coincident ndpoints.Hencewemay assume haty is adjacent o some elementof A, and so G-{f, x} is connected. C1

THEOREM 1. If G is a connected graph of even order, then G2 has a1-factor.PROOF. Noting that the resultholds for graphsof low order, we pro-ceed by induction.Suppose hat G is connectedof even orderand that thetheoremholdsforgraphsof smaller rder.ByLemma1wemayfindpointsx, y E G such that d(x,y)_2 and G-{x,y} is connected(for we mayeither choose x andy to be coincidentendpointsor, if none such exist,we may. ake x andy to be the points guaranteedby Lemma1). Thus(G {x, y})2 has a 1-factor,whichtogetherwith the edgexy of G2yieldsa 1-factor or G2. 0Thus since the total graph T(G)of a graphG satisfiesT(G)= [S(G)]2(Behzad [2])whereS(G)is the subdivision raphof G, we haveCOROLLARY 1. Every connected,totalgraph of even orderhas a 1-factor.THEOREM . If G is a connectedgraph of order 2n (n>1) and k is an

integer, 1


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10 D. P. SUMNER [January

COROLLARY. If G is a connectedgraph of even order with no inducedK1,3,then G has a 1-factor.PROOF. The onlyconnectedgraphof orderfourwhichdoes not havea 1-factor s K13. Thus (except in the trivial case whereG is K2) thecorollary s the specialcase of the theoremwith k=2. DSinceno linegraphcancontainK1,3 s an induced ubgraph,we obtain

COROLLARY. Every connected inegraph of even order has a 1-factor.A 1-factorof a graphmay be viewedas a partitionof the pointsintotwo-elementsubsets in such a way that the elements of each set areadjacent.As a result of Corollary3, we see that it is always possible toforma similarpartitionof the edgesaslongas thereare aneven number fthem. Similarly,as a result of Corollary1, if a graphcontainsan evennumberof elements(i.e., points and lines), then we may partitiontheelements nto subsetsof order wo so that the elementsof each subsetareeitheradjacentor incident.As another mmediate onsequence f Corollary2 we haveCOROLLARY. Every connected cubic graph in which every point lies

in a triangle has a 1-factor.REMARK. An alternateproof of Corollary is obtainedby notingthatin any such graphotherthan K4 he collectionof those edgesthat lie in aneven numberof triangles onstitutesa 1-factor.As an example,considerthe graph in Figure 1. The edges whichare numbereddetermine uch a1-factor.

2 51 ~~~47

FIGURE 1Ournext result s anextensionof Theorem2. By an end-blockwe meanone which containsexactlyone cutpoint.THEOREM. Let G be a connectedgraph of even order.If(i) G has exactly one block of even order, and(ii) for each block B of G, there exists an integer k, 3


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1974] GRAPHS WITH 1-FACTORS 11PROOF. If G hasexactlyone block,thenG hasa 1-factorbyTheorem2. SupposeG is a graphsatisfyinghe conditions i) and (ii) suchthatthe

theoremholdsfor graphswith fewerblocks.SinceG hasat leasttwoend-blocks,thereexistsan end-blockB of odd order.Letb EB be the uniquecutpointof Glyingin B, andlet k be the integerassociatedwith B. ThenB-{b} is connectedof evenorderandevery nduced,connected ubgraphof B-{b} havingorderk has a 1-factor notek< IBI-1 sinceIBI s odd).Thus B-{b} has a 1-factorby Theorem1. Also (G-B)U{b} has fewerblocksthanGand satisfies i) and (ii)andhencehas a 1-factor.Thusweobtaina 1-factor or G. DThenexttheoremmay be proven n a similarmanner.THEOREM . If G is a connectedgraph of even ordersuch that(i) G has exactly one block of even order, and(ii) no block of G contains an inducedK1,3,then G has a 1-factor.

THEOREM5. If G is a connectedgraph of even order and every inducedK1,3 ontainsa bridgewhose deletionresultsin twocomponentsof even order,then G has a 1-factor.

PROOF.Weinducton the numberof points n Gnotingthat the resultholdsfor small orders.Let G be a graphsatisfying he hypothesesof thetheoremand such hatthetheoremholdsforgraphswithfewerpoints hanG. If G has no inducedK13, then G has a 1-factorby Corollary2. If GdoesnotcontainaninducedK1,3,et e=ab be anedgeof thisK1,3uchthatthe deletionof e resultsn a graphhaving hetwoevencomponentsA andB witha EA andb EB. SupposeA containsan inducedK1,3.Letf be anedgeof thissubgraphuch thatG-{f} contains wo componentsof evenorder.Since both endpointsoff lie in A, A-{f} consists of two com-ponentsC and D. Wemayassumea E D. Then C isoneof thecomponentsof G-{f} and hencehasevenorder(asmust D also).ThusA satisfiesheconditionsof the theoremand so musthave a 1-factor.Similarly,B con-tains a 1-factorwhichtogetherwith that for A producesa 1-factor orG. ESincethis paperwas submitted,we have learned hat our Theorem1,Corollary 1, and Corollary 3 were discoveredindependentlyby G.Chartrand,A. Polimeni,and J. Stewartand occur in their paper Theexistence of a 1-factor in line graphs, squares, and total graphs, which isto appear n IndagationesMathematicae.

ACKNOWLEDGMENT. Thanksare dueto FrankR. Bernhartorpointingout to me an error n a previousversionof thispaper.


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12 D. P. SUMNERREFERENCES

1. L. W. Beinekeand M. D. Plummer,Onthe 1-factorsof a nonseparableraph,J.CombinatorialTheory2 (1967),285-289. MR 35 1499.2. M. Behzad, A criterionor theplanarityof the totalgraphof a graph, Proc. Cam-bridgePhilos. Soc. 63 (1967),679-681. MR 35 2771.3. F. Harary,Graph heory,Addison-Wesley,Reading,Mass., 1969. MR 41 1566.4. J. Petersen,Die TheoriederreguldrenGraphen,Acta Math. (1891), 193-220.5.,W. T. Tutte, The actorizationsof lineargraphs,J. LondonMath. Soc. 22 (1947),107-111. MR 9, 297.DEPARTMENTOF MATHEMATICS,UNIVERSrrY OF SOUTH CAROLINA, COLUMBIA, SOUTH

CAROLINA 29208

graphs with 1-factors

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