Randerath, Bert

Vestergaard, Preben D.

English

    Aalborg UniversitetAll P3-equipackable graphsRanderath, Bert; Vestergaard, Preben D.Publication date:2008Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Randerath, B., & Vestergaard, P. D. (2008). All P3-equipackable graphs. Department of Mathematical Sciences,Aalborg University. Research Report Series No. R-2008-10General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: November 28, 2020AALBORG UNIVERSITY'&$%All P3-equipackable graphsbyBert Randerath and Preben Dahl VestergaardTo appear in Discrete MathematicsR-2008-10 October 2008Department of Mathematical SciencesAalborg UniversityFredrik Bajers Vej 7G DK - 9220 Aalborg Øst DenmarkPhone: +45 99 40 80 80 Telefax: +45 98 15 81 29URL: http://www.math.aau.dk eISSN 1399–2503 On-line version ISSN 1601–7811All P3-equipackable graphsBert RanderathInstitut für InformatikUniversität zu KölnD-50969 Köln, Germanyranderath@informatik.uni-koeln.dePreben Dahl VestergaardDepartment of Mathematical SciencesAalborg UniversityDK-9220 Aalborg ∅, Denmarkpdv@math.aau.dkOctober 7, 2008Accepted for publication in Discrete MathematicsAbstractA graph G is P3-equipackable if any sequence of successive removals of edge-disjoint copiesof P3 from G always terminates with a graph having at most one edge. All P3-equipackablegraphs are characterised. They belong to a small number of families listed here.Keywords: Packing, equipackable, randomly packable, covering, factor, decomposition, equire-movable2000 Mathematics Subject Classification: 05C70, 05C351 IntroductionLet H be a subgraph of a graph G. An H-packing in G is a partition of the edges of G intodisjoint sets, each of which is the edge set of a subgraph of G isomorphic to H, and possibly aremainder set. For short, E(G) is partitioned into copies of H and maybe a remainder set. Welist some references to an extensive literature at the back. A graph is called H-packable if G isthe union of edge disjoint copies of H. An H-packing is maximal if the remainder set of edgesis empty or contains no copy of H. An H-packing is maximum if E(G) has been partitionedinto a maximum number of sets isomorphic to H and a possible remainder set. A graph iscalled H-equipackable if any maximal H-packing is also a maximum H-packing, i.e., a graph Gis H-equipackable if successive removals of copies of H from G can be done the same numberof times regardless of the particular choices of edge sets for H in each step. If every maximalH-packing of a graph G uses all edges of G, then G is called randomly H-packable. Equivalently,G is randomly H-packable if each H-packing can be extended to an H-packing containing alledges of G, see e.g. [1, 2, 5, 6].Zhang and Fan [9] have studied H-equipackable graphs for the case H = 2K2. We shall considerpath packing and in the following H will always be assumed to be the graph P3, the path oflength two, and we may omit explicit reference to it. A graph G is then (P3-) equipackableif successive removals of two adjacent edges from G can be done the same number of timesregardless of the particular choices of edge pairs in each step. A component consisting of onevertex is called trivial, a non-trivial component contains an edge. A graph has order |V (G)| andsize |E(G)|. A graph of odd (even) size is called odd (even). A vertex of valency one is called aleaf. A star is called even if its size is even, and by K1,2k we denote the even star with 2k leaves.Observation 1 A graph is randomly H-packable if and only if it is H-packable and H-equipackable.S. Ruiz [7] characterised randomly P3-packable graphs.Theorem 2 A connected graph G is randomly packable if and only if G ∼= C4 or G ∼= K1,2k,k ≥ 1.Y. Caro, J. Schönheim [3] and S. Ruiz [7] stated the following result.Lemma 3 A connected graph is packable if and only if it has even size.This immediately implies Corollary 4 below.Corollary 4 If a connected graph is equipackable, a maximal packing either contains all edgesor all but one edge of the graph.From B.L. Hartnell, P.D. Vestergaard [4] and P.D. Vestergaard [8] we have the following obser-vation.Observation 5 Let G be an equipackable graph. Then any sequence of P3-removals from G willproduce an equipackable graph.From Corollary 4 and Observation 5 we obtainCorollary 6 Let G be a connected graph. If there is a sequence of P3-removals from G thatcreates more than one component of odd size, then G is not equipackable.We now state our main result, a characterisation of all equipackable graphs with at most onenon-trivial component:Theorem 7 Let G = (V, E) be a graph with at most one non-trivial component. Then G isequipackable if and only if its non-trivial component belongs to one of the thirteen families listedin Figure 1 or can be obtained by a sequence of P3-removals from such a graph.Clearly, we wish those thirteen families listed to be maximal w.r.t. P3-removals, i.e., no graphfrom one of the families can be obtained as a subgraph of a larger equipackable graph by removinga P3 from it.In the figures below we indicate by an arrow from which family of graphs we may obtain thegiven graph by a sequence of P3-deletions. The shaded vertex sets may vary in cardinality.213F14F3F 5F2FK  −  e 58F7FF610FF11F12F9F# eveneven stars7−circuit# odd# even# odd# odd # even # odd# odd(# even) (# even)Figure 1: All connected, maximal with respect to P −removal, P −equipackable graphs3 3# odd4−circuitWe will prove this characterisation in the following section.2 Proof of Theorem 7By Lemma 3 and Theorem 2 a graph with at most one non-trivial component, which has evensize, is equipackable if and only if its non-trivial component is a 4-circuit or an even star (Fig-ure 2). Thus it only remains to characterise equipackable graphs with at most one non-trivialcomponent of odd size.34−circuit even stars# evenFigure 2:   Connected P −equipackable graphs of even size (Ruiz graphs)  2FF13In [8] P.D. Vestergaard examined equipackable graph with all degrees ≥ 2 and stated the fol-lowing result.Theorem 8 A connected graph G with all degrees ≥ 2 is equipackable if and only if G is one ofthe graphs listed in Figure 3.4F3F 5FFF1 F6F12FF 999 F93Figure 3: All connected P −equipackable graphs G without leavesObserve that this solution contributes to our characterisation five graphs (F6, F3, F4, F5, F9)maximal with respect to P3-removals. All other graphs of this solution are obtained by a se-quence of P3-removals from graphs of the thirteen graph families of our characterisation. Thus itnow remains to characterise equipackable graphs G which have only one non-trivial component,say H, where H has odd size and contains a leaf.Since H has a leaf, it also has a bridge. Let b = xy be a bridge of H. Throughout we shalldenote the two components of H −xy by H1 and H2 with x ∈ V (H1), y ∈ V (H2). We shall firsttreat the case that G has a non-leaf bridge, then the case that all bridges are leaf bridges.Case 1: Assume b = xy is a non-leaf bridge of G, i.e., deg(x) ≥ 2, deg(y) ≥ 2.Subcase 1.1: Assume further that H has a maximum P3-packing P which does not contain b.Since P by Corollary 4 contains all but one edge of G and b /∈ P, we have for i = 1, 2 that P∩Hiis a P3-packing of Hi and therefore Hi has even size ≥ 2.Let z ∈ N(x) \ {y}. By P3-removal of zxy we obtain an equipackable graph which has an oddcomponent contained in H1 − xz, and H − {zx, xy} also has the even component H2 which isconnected, randomly packable and hence, by Observation 1, is either a 4-circuit or an even star.By symmetry also H1 is a 4-circuit or an even star. Therefore H belongs to one of the familiesof graphs depicted in Figure 4.48F7FF612F10FF12# odd# even# even # even# odd # oddFigure 4:  Connected, P −equipackable graphs in Case 1.13# odd # evenx       yx       y x       yx       yx                         yx       yNote that only three new graph families (F7, F8, F10) maximal with respect to P3-removals con-tribute in this case to our characterisation. All other graph families of this subcase are obtainedby a sequence of P3-removals from graphs of the thirteen graph families of our characterisation.Subcase 1.2: Assume now that each non-leaf bridge of H is contained in every maximumP3-packing.With notation as above let b = xy be a non-leaf bridge of H, the components of H − xy areH1, H2. Their sizes have the same parity since H has odd size. If H1, H2 both had even sizethey would be P3-packable and H would have a maximum P3-packing not containing b in con-tradiction to assumption. Therefore H1, H2 both have odd size.Claim: At least one of H1, H2 is an odd star.Proof. P3-removal from H of zxy, z ∈ N(x) \ {y}, creates an odd size component, namely H2.If H2 is an odd star we are finished. Otherwise, we can isolate an odd component inside H2: IfdegH2(y) is even we P3-remove all edges incident to y in pairs and if degH2 is odd we P3-removeall but one edge incident to y in pairs and that remaining edge yw, w ∈ N(y), together withwr, r ∈ N(w) \ {y} (Since H2 is not an odd star there has to exist at least one such vertex w).Then H1∪{xy} is even, connected, randomly packable and hence is either a 4-circuit or an evenstar. Since H1∪{xy} contains a leaf, it is an even star and hence H1 is an odd star. That provesthe claim.2Suppose H1 and H2 are both odd stars. Now assume that, say x, is not the center of H1and let v be the center of H1. Since vx is a non-leaf bridge and there obviously exists a maxi-mum P3-packing P which does not contain vx, we obtain a contradiction to the assumption ofSubcase 1.2. Hence we find that H is obtained by adding an edge between the centers of H1and H2 (see Figure 5). Consequently H can be obtained from one of the graphs of the familyF12 in our characterisation by P3-deletions.5If, say, H2 is an odd star and H1 is not, then P3-removal of zxy from H, z ∈ N(x) \ {y}, givesthat H1 − xz has even size.Now assume that zx is a leaf bridge of H (and likewise of H1), i.e., degH(z) = 1.Then P3-removal of zxy leaves the odd component H2 and H1 − xz with one non-trivial evencomponent. Thus the non-trivial even component of H1 − xz is either a 4-circuit or an evenstar. The former yields easily a non-equipackable graph, the latter gives that H1 is an odd star,a contradiction to assumption on H1.Suppose now that zx is a non-leaf bridge of H (and likewise for H1).The two components of H1 − xz have sizes of same parity. That cannot be odd since G − zxywould then have three odd components in contradiction to Corollary 6. It cannot be even eitherbecause then we could easily construct a maximum P3-packing P which does not contain thenon-leaf bridge xz, a contradiction to the basic assumption of this subcase.So we may for all z ∈ N(x) \ {y} assume that xz is not a bridge of H (and H1).P3-removal of zxy for z ∈ N(x) \ {y} produces the connected, even component H1−xz which isthen randomly P3-packable and hence is either an even star or a 4-circuit. If H1−xz is a 4-circuitwe are immediately led to H not being equipackable because, if a, b, c, d are the edges of this4-circuit (in cyclic order) then the packing {xy, a}, {xz, c} cannot be extended to a maximumpacking of H. Observe that we have N(x) \ {y} = {z1, z2, ..., zp} with z1 = z and p ≥ 2. Thusfor all zi ∈ N(x) \ {y} the connected graph H1 − xzi is an even star.It follows that p = 2 andH1−xzi must always be isomorphic to a P3 = K1,2 with a center vertex z3−i having neighboursx and zi for i = 1, 2. Thus H1 is a 3-circuit with vertices x, z1, z2 with x joined to y, and y hasan odd number of leaves attached (see Figure 5).F13F12# odd # odd # oddFigure 5: Connected P −equipackable graphs in Case 1.2 3Observe that none of these equipackable graph families are new families maximal with respectto P3-removals for our characterisation. Both graph families of this subcase are obtained by asequence of P3-removals from graphs of the graph families (F12, F13) of our characterisation. Wemay now assume that there exist no non-leaf bridge of H.Case 2: All bridges of H are leaf bridges and there exists at least one bridge b = xy ofH, i.e. H2 = {y}.If all xz, z ∈ N(x) \ {y}, are bridges of H, then they are leaf bridges and H is an odd star,derivable from a member of our characterisation by a sequence of P3-removals. Thus we mayassume that x is contained in at least one cycle of H1 and there exist at least two edges incidentto x, which are not bridges.If x has an even number of neighbours in H1 we can isolate xy by pairing up and P3-removingall xz, z ∈ N(x)\{y}. If x has an odd number of neighbours in H1 we isolate xy by P3-removing6all xz, z ∈ N(x) \ {y}, and one further edge zw, w ∈ N(z) \ {x} (observe that such an edgehas to exist). For simplicity, let E′ be the set of edges of all P3’s necessary to remove in orderto isolate the bridge xy and H ′ = H − E′. Since xy is isolated in H ′ and H ′ is equipackable,we obtain by Lemmas 3, Observation 5 and Corollaries 4, 6 that all non-trivial components Dnot containing x and y are randomly packable and therefore of even size ≥ 2. Thus every suchnon-trivial component D is either a 4-circuit or an even star.Assume that one of these components is a 4-circuit C with vertices {c0, c1, c2, c3} and edges{cic(i+1)mod4|0 ≤ i ≤ 3}.As all bridges of H are leaf bridges, with EC = {xc|c ∈ NH1(x) ∩ V (C)} we have |EC | ≥ 2.It is easy to see that we can remove two (if |E(C)| = 2) or three P ′3s from the subgraph of Hinduced by {x} ∪ V (C) to produce two (if |E(C)| = 3) or three isolated edges (including xy) incontradiction with Corollary 6.If |EC | = 2 there exist i, j, k, ℓ = {0, 1, 2, 3} such that xci, xcj ∈ EC and xcick, xcjcℓ are P3’s ofH that isolate the two independent edges cick, cjcℓ remaining in C. By Corollary 6 then H isnot equipackable, a contradiction. If |EC | = 3, without loss of generality we may assume thatEC = {xc0, xc1, xc2} and in that case E′∪{c0c3, c1c2} is an edge set of even size, which can pairedup in P ′3s whose removal isolate two edges c0c1 and c2c3 on C, by Corollary 6 that contradictsH being equipackable. If |EC | = 4, again E′ ∪ {c0c3, c1c2} has even size and can be paired upand P3-removed to leave two independent edges c0c1 and c2c3 on C, giving a contradiction toH being equipackable.Hence every such non-trivial component D not containing x and y is an even star.Now suppose there exist two different components R1 and R2 of this kind. Analogously to theprevious argumentation let ERi be the subset of E′ of edges incident to the vertices of Ri fori = 1, 2. Since H is connected, and all bridges of H are leaf bridges there has to exist for eachi = 1, 2 at least two edges f ′i , f ′′i of ERi adjacent to an edge of Ri. Pairing up f ′i with one edge ofE(Ri), say fi, i = 1, 2, and P3-removing all remaining edges of E′ (their number is even, recallthat fi /∈ E′) will isolate two odd stars ER1 − f1 and ER2 − f2, a contradiction to Corollary 6.Thus there exists only one non-trivial component R of H ′ not containing x and y, and that isan even star.We now distinguish between two cases depending on the parity of degH1(x). Assume degH1(x)is even. Then obviously H is, regardless of whether the centre r of R is adjacent to x or not, amember of the graph family F12 or can be obtained by a sequence of P3-removals from a memberof F12.Now it remains to consider that degH1(x) is odd, i. e. dH(x) is even. As already noted at thebeginning of Case 2 the vertex x must be contained in at least one cycle of H1 and there existat least two edges incident to x, which are not bridges. Since R is an even star K1,2l with l ≥ 1it is not difficult to deduce that the cycle has length ≤ 5. First let R be a star with at least fourbranches. Recall that E′ is the set of edges of all P3’s necessary to remove in order to isolatethe bridge xy and let H ′ = H − E′. Moreover, since x has an odd number of neighbours in H1we isolate xy by P3-removing all xz, z ∈ N(x) \ {y}, and one further edge zw, w ∈ N(z) \ {x}.Regardless of the choice of this additional edge zw the remainder will be an even star with atleast four edges. Concatenation of all ingredients builds up a member of F12 or a graph that canbe obtained by a sequence of P3-removals from a member of F12. Therefore we conclude that Ris always an even star with two branches regardless of the choice of the additional edge zw. By7inspection we obtain that H is either the graph F11 or F13 depicted in Figure 6.F11F13F12# even # odd # odd(# even) (# even)# odd Figure 6: Connected P −equipackable graphs in Case 23This completes the proof of our main result. 2The proof can also be done by induction on |E(G)|, but the arguments are not shorter.Acknowledgement: The authors thank the anonymous referees for careful reading and valu-able comments on this text.References[1] L.W. Beineke, P. Hamburger, W.D. Goddard, Random packings of graphs, Discrete Math. 125(1994),45-54.[2] J. Bosák, Decompositions of graphs, Kluwer Academic Publishers, Dordrecht, (1990).[3] Y. Caro, J. Schönheim, Decompositions of trees into isomorphic subtrees, Ars Combin. 9 (1980),119-130.[4] B.L. Hartnell, P.D. Vestergaard, Equipackable graphs, Bull. Inst. Combin. Appl. 46 (2006), 35-48.[5] R. Molina, On randomly Pk-decomposable graphs, Congr. Numer. 148 (2001), 207-221.[6] R. Molina, M. McNally and K. Smith, Characterising randomly Pk-decomposable graphs for k≤ 9,Congr. Numer. 156 (2002), 211-221.[7] S. Ruiz, Randomly decomposable graphs, Discrete Math. 57 (1985), 123-128.[8] P.D. Vestergaard, A short update on equipackable graphs, Discrete Math. 308 (2008), 161-165.[9] Y. Zhang, Y. Fan, M2-equipackable graphs, Discrete Appl. Math. 154 (2006), 1766-1770.8

All <em>P<sub>3</sub></em>-equipackable graphs

https://vbn.aau.dk/ws/files/15001360/R-2008-10.pdf

All <em>P<sub>3</sub></em>-equipackable graphs

Abstract

Similar works

Full text

Available Versions

VBN