Lyngsie, Kasper Szabo

Merker, Martin

We prove that every connected graph can be edge-decomposed into a spanning tree, an even graph, and a star forest

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Downloaded from orbit.dtu.dk on: May 15, 2019Decomposing graphs into a spanning tree, an even graph, and a star forestLyngsie, Kasper Szabo; Merker, MartinPublished in:The Electronic Journal of CombinatoricsPublication date:2019Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Lyngsie, K. S., & Merker, M. (2019). Decomposing graphs into a spanning tree, an even graph, and a star forest.The Electronic Journal of Combinatorics, 26(1), [P1.33].Decomposing graphs into a spanning tree,an even graph, and a star forestKasper Szabo Lyngsie Martin Merker∗Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngby, Denmark{ksly,marmer}@dtu.dkSubmitted: Jun 29, 2018; Accepted: Feb 21, 2019; Published: Mar 8, 2019c©The authors. Released under the CC BY-ND license (International 4.0).AbstractWe prove that every connected graph can be edge-decomposed into a spanningtree, an even graph, and a star forest.Mathematics Subject Classifications: 05C05, 05C381 IntroductionAll graphs in this paper are simple and finite. A decomposition of a graph G is a collectionof edge-disjoint subgraphs whose union is G. A graph is called even if every vertexhas even degree. It is easy to see that every graph can be decomposed into a forestand an even graph. In 1979, Malkevitch [9] studied cubic graphs which admit such adecomposition where the forest is a spanning tree. In this case it is equivalent to theexistence of a spanning tree containing no vertices of degree 2. Such a spanning tree iscalled homeomorphically irreducible, or a HIST.For general graphs, the existence of a HIST is much less restrictive than the existenceof a decomposition into a spanning tree and an even graph. However, even the existence ofa HIST is not guaranteed by large connectivity or regularity, as was shown by Albertsonet al. [3]. Douglas [5] showed that it is NP-complete to decide whether a planar subcubicgraph contains a HIST.In cubic graphs, the removal of the edges of a spanning tree results in a collection ofcycles and paths. Hoffmann-Ostenhof [4] (see also [6]) conjectured that the spanning treecan be chosen such that the collection of paths is a matching.∗Supported by the Danish Council for Independent Research, Natural Sciences, grant DFF-1323-00178,AlgoDisc.the electronic journal of combinatorics 26(1) (2019), #P1.33 1Conjecture 1 (3-Decomposition Conjecture). Every connected cubic graph can be de-composed into a spanning tree, a collection of cycles, and a matching.Akbari, Jensen, and Siggers [2] showed that any cubic graph has a decomposition intoa spanning forest, a collection of cycles, and a matching. Abdolhosseini et al. [1] verifiedthe 3-Decomposition Conjecture for traceable cubic graphs and Ozeki and Ye [10] verifiedit for 3-connected planar cubic graphs. The latter was extended by Hoffmann-Ostenhof,Kaiser, and Ozeki [7] to all planar cubic graphs. The following theorem is the main resultof our paper and is in some sense a generalization of the 3-Decomposition Conjecture tothe class of all connected graphs.Theorem 2. Every connected graph can be decomposed into a spanning tree, an evensubgraph, and a star forest.As a special case our result implies that every cubic graph has a decomposition intoa spanning tree, a collection of cycles, and a collection of paths of length at most 2. Thiswas previously shown by Li and Cui [8].One might be tempted to think that every connected graph admits a decompositioninto a spanning tree, an even graph, and a matching. However, this is easily seen tobe false since the complete bipartite graph K2,n has no such decomposition. Such adecomposition is also not guaranteed if we restrict our attention to regular graphs.Theorem 3. For each r > 4, there exists an r-regular connected graph which has nodecomposition into a spanning tree, an even graph, and a matching.Proof. Let r > 4 be given and let G be the graph obtained from Kr+1 by subdividingeach edge once. Let G′ be a graph obtained from Kr+1 by subdividing r − 2 edges onceand adding an edge between each pair of vertices of degree 2. For each vertex v of degree2 in G, let Gv denote a copy of G′. Now let G′′ be obtained from the disjoint union of Gand all the graphs Gv by adding edges between v and the vertices of degree r − 1 in Gv,for each vertex v of degree 2 in G. Note that G′′ is r-regular and any decomposition of G′′into a spanning tree, an even graph, and a matching also induces such a decompositionof G. Clearly, the even graph cannot contain any edges of G, therefore this correspondsto a decomposition of G into a spanning tree and a matching. The graph G has r(r + 1)edges, and every spanning tree of G has r+ r(r+1)2edges, thus the matching has to containat least r(r−1)2> r + 2 edges. However, the size of a maximal matching in G is r + 1, soG cannot be decomposed into a spanning tree and a matching.The construction in the proof above shows that for r-regular graphs the size of thestars in the forest in Theorem 2 grows at least linearly in r.2 Proof of Theorem 2Before we begin the proof of our main theorem, we introduce a few definitions.the electronic journal of combinatorics 26(1) (2019), #P1.33 2Definition 4 (separating cycle). A cycle C in a connected graph G is called separatingif G− E(C) is disconnected.Note that in the literature a cycle is called separating in a graph if the removal ofits vertex-set results in a disconnected graph, while for us the removal of its edge-set isrelevant. In particular, every cycle containing a vertex of degree 2 is separating.Definition 5 (fragile). A graph G is called fragile, if G is connected and every cycle ofG is separating.Fragile graphs have also been investigated in the context of planar graphs by Hoffmann-Ostenhof et al. [7]. Their 2-Decomposition Conjecture, which is equivalent to Conjec-ture 1, states that every subcubic fragile graph can be decomposed into a spanning treeand a matching.Note that if we remove an even subgraph H such that G− E(H) is connected and if wechoose such a subgraph of maximal size, then G − E(H) is fragile. In particular, everyconnected graph decomposes into an even graph and a fragile graph. Therefore it is suf-ficient to prove that every fragile graph has a decomposition into a spanning tree and astar forest. For brevity, we introduce the following notation.Definition 6 (starlit). A spanning tree T of a graph G is called starlit if G−E(T ) is astar forest.All we need to show is that every fragile graph contains a starlit spanning tree. Weprove an even stronger result, where we prescribe that all edges at a specified vertexbelong to the spanning tree in the decomposition.Definition 7 (v-full). A spanning tree T of a graph G is called v-full for some vertex vin G, if all edges incident with v in G are also in T .We can now state the theorem we are going to prove.Theorem 8. If v is a vertex in a fragile graph G, then G has a starlit v-full spanningtree.As already discussed, Theorem 2 follows immediately from Theorem 8. We finish thissection by proving Theorem 8.Proof of Theorem 8. Let G be a counterexample of minimal size.Claim 1: G is 2-connected.Proof. Suppose the claim is false and u is a cutvertex in G. Let K be a component ofG − u, let G1 denote the subgraph of G induced by V (K) ∪ {u}, and let G2 denote thegraph induced by the edges in G − E(G1). We can assume that v ∈ V (G1). Clearly G1and G2 are fragile and contain fewer edges than G, so G1 contains a starlit v-full spanningtree T1, and G2 contains a starlit u-full spanning tree T2. Now the union of T1 and T2 isa starlit v-full spanning tree in G.the electronic journal of combinatorics 26(1) (2019), #P1.33 3Note that Claim 1 implies that the minimum degree of G is at least 2.Claim 2: There are no adjacent vertices of degree 2 in G.Proof. Suppose x and y are two adjacent vertices of degree 2 and let z denote the neighbourof y different from x. We may assume without loss of generality that v 6= y. The graphG′ = G− xy is fragile, so by minimality of G there exists a starlit v-full spanning tree T ′of G′. If v 6= x, then T = T ′ is also a starlit v-full spanning tree of G. If v = x, then wechoose instead a starlit z-full spanning tree T ′′ of G′. Now T ′′ + xy− yz is a starlit v-fullspanning tree of G.Let H be the subgraph of G induced by the vertices of degree at least 3.Claim 3: H contains no isolated vertices and no cycles of length 3.Proof. Suppose u is an isolated vertex in H. That is, u is a vertex of degree at least 3 inG all of whose neighbours have degree 2.First, suppose u = v. Let x be a neighbour of u, and y the neighbour of x different fromu. By Claim 2, y has degree at least 3 and is therefore not adjacent to u. Let G′ bethe graph obtained from G by removing x and adding the edge uy. Since u has only oneneighbour of degree greater than 2 in G′, every cycle through u is still separating. Thus,G′ is fragile and contains a starlit u-full spanning tree T ′. Now T = T ′ − uy + ux+ xy isa starlit u-full spanning tree of G.Thus we can assume u 6= v. The graph G′ = G− u is connected by Claim 1. Clearly G′is fragile and therefore contains a v-full starlit spanning tree T ′. If v is a neighbour of u,then T = T ′ + uv is a starlit v-full spanning tree of G. If v is not a neighbour of u, thenadding an arbitrary edge incident with u to T ′ results in a starlit v-full spanning tree ofG. This contradiction shows that the minimum degree of H is at least 1.Finally, suppose H contains a cycle C of length 3. Since every vertex of C has degreeat least 3, and since G is 2-connected, it is easy to see that C is not separating, whichcontradicts G being fragile.For u ∈ V (H), we write dH(u) to denote the degree of u in H and dG(u) for its degreein G.Claim 4: If u is a vertex in H different from v, then dH(u) > 2.Proof. Suppose u is a vertex of degree 1 in H, u 6= v, and x is the neighbour of u in H.Let G′ = G − u. First, suppose that v is not a degree 2 vertex adjacent to u in G. ByClaim 1, the graph G′ is fragile, so it has a starlit v-full spanning tree T ′ by minimalityof G. Now T = T ′+ ux is a v-full starlit spanning tree of G. Thus, we can assume that vhas degree 2 and is a neighbour of u in G. Let T ′′ be a starlit x-full spanning tree of G′.Clearly T = T ′′ + uv is a v-full spanning tree of G. Since T ′′ is x-full, the spanning treeT is also starlit, contradicting our choice of G.the electronic journal of combinatorics 26(1) (2019), #P1.33 4Claim 4 implies that there exists a cycle in H. The following claim shows that thereare at most two vertices in H which have degree less than 3 in H.Claim 5: If u ∈ V (H) and dH(u) = 2, then either u = v or dG(v) = 2 and uv ∈ E(G).Proof. Suppose u is a vertex of degree 2 in H, u 6= v, and dG(v) > 3 or uv /∈ E(G). Letx and y denote the neighbours of u in H. Note that all other neighbours of u in G havedegree 2. Let G′ be the graph obtained from G−u by adding the edge xy. Claim 1 impliesthat G is connected and Claim 3 implies that G′ has no multiple edges. For a cycle C ′ inG′ containing xy, the corresponding cycle C in G, which is obtained from C ′ by replacingxy with the path xuy, is separating if and only if C ′ is separating. Thus, G′ is fragile andcontains a starlit v-full spanning tree T ′. If xy ∈ E(T ′), then T = T ′ − xy + ux+ uy is astarlit v-full spanning tree in G. Thus, we can assume xy /∈ E(T ′). Since G′ −E(T ′) is astar forest, at least one of x and y has degree 1 in G′ − E(T ′), say x. Now T = T ′ + uyis a starlit v-full spanning tree in G.Let C be a cycle in H for which the component of G − E(C) containing v has max-imal size. Note that C is induced. Let K denote the component of G−E(C) containing v.Claim 6: H ⊆ K ∪ C.Proof. Since C is chordless, it suffices to show V (H) ⊆ V (K ∪C). Suppose u is a vertexin H which is not in K or C. Let L denote the component of G − E(C) containingu. There exists no cycle in L ∩ H since that cycle would contradict the choice of C.Claim 4 now implies that L contains a path P joining two vertices a and b on C suchthat all intermediate vertices are in V (H) \V (C). Let P1 and P2 be the two edge-disjointsubpaths of C joining a and b. We may assume that P2 contains a vertex of K. Now thecycle formed by the union of P and P1 contradicts the choice of C.Since G is fragile, the graph G − E(C) is disconnected so there is a vertex u on Cwhich is not in K. Since C is induced, the vertex u has exactly two neighbours on C.Claim 6 implies that all neighbours of u not on C have degree 2. Now dH(u) = 2 andu 6= v. By Claim 5, we have dG(v) = 2 and uv ∈ E(G), which implies that u is in K,contradicting our choice of u.References[1] F. Abdolhosseini, S. Akbari, H. Hashemi, and M. Moradian, Hoffmann-Ostenhof’s conjecture for traceable cubic graphs, arXiv:1607.04768, (2016).[2] S. Akbari, T. R. Jensen, and M. Siggers, Decompositions of graphs into trees,forests, and regular subgraphs, Discrete Mathematics, 338 (2015), pp. 1322 – 1327.[3] M. O. Albertson, D. M. Berman, J. P. Hutchinson, and C. Thomassen,Graphs with homeomorphically irreducible spanning trees, Journal of Graph Theory,14 (1990), pp. 247–258.the electronic journal of combinatorics 26(1) (2019), #P1.33 5[4] P. J. Cameron, Research problems from the BCC22, Discrete Mathematics, 311(2011), pp. 1074 – 1083.[5] R. J. Douglas, NP-completeness and degree restricted spanning trees, DiscreteMathematics, 105 (1992), pp. 41 – 47.[6] A. Hoffmann-Ostenhof, Nowhere-zero flows and structures in cubic graphs, PhDthesis, University of Vienna, 2011.[7] A. Hoffmann-Ostenhof, T. Kaiser, and K. Ozeki, Decomposing planar cubicgraphs, Journal of Graph Theory, 88 (2018), pp. 613–640.[8] R. Li, and Q. Cui, Spanning trees in subcubic graphs, Ars Combinatoria, 117(2014), pp. 411–415.[9] J. Malkevitch, Spanning trees in polytopal graphs, Annals of the New YorkAcademy of Sciences, 319 (1979), pp. 362–367.[10] K. Ozeki and D. Ye, Decomposing plane cubic graphs, European Journal of Com-binatorics, 52 (2016), pp. 40 – 46.the electronic journal of combinatorics 26(1) (2019), #P1.33 6

Decomposing graphs into a spanning tree, an even graph, and a star forest

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Decomposing graphs into a spanning tree, an even graph, and a star forest

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