Shortest cycle covers and cycle double covers with large 2-regular subgraphs
SShortest cycle covers and cycle double covers withlarge 2-regular subgraphs
Jonas H¨agglund, Klas Markstr¨omMay 18, 2018
Abstract
In this paper we show that many snarks have shortest cycle coversof length m + c for a constant c , where m is the number of edgesin the graph, in agreement with the conjecture that all snarks haveshortest cycle covers of length m + o ( m ).In particular we prove that graphs with perfect matching index atmost 4 have cycle covers of length m and satisfy the (1 , m . We also prove some results for graphswith low oddness and discuss the connection with Jaeger’s Petersencolouring conjecture. A cycle cover of a graph G is a collection of cycles F such that every edgeedge of G belongs to at least one cycle of F . Cycle covers are well studiedobjects, see [Zha12, Jac93] for surveys, and in particular the length, i.e thesum of the lengths of the cycles in the cover, has received much attention.We let scc ( G ) denote the length of the shortest cycle cover of G . In [AT85]Alon and Tarsi made the following conjecture Conjecture 1.1. [AT85] If G is a cubic 2-edge connect graph then scc ( G ) ≤ m , where m is the number of edges in G If G is 3-edge-colorable it is easy to see that the conjecture is true, andin fact scc ( G ) = m , since we can use the cycle cover given by taking twopairs of edge colours. This particular covering will also be a (1 , , , a r X i v : . [ m a t h . C O ] J un pen, it is known from [AT85] that for both 2-edge and 3-edge-connectedgraphs the constant would be optimal, and the best current bound is scc ( G ) ≤ m , [KKL + n ≤
36 vertices were generated and, among other things,the value of scc ( G ) was computed. It turns out that in this collection ofgraphs only two have scc ( G ) = m + 1, the examples are the Petersen graphand a particular snark with 34 vertices, while all other have scc ( G ) = m ,and the authors made the following conjecture. Conjecture 1.2. [BGHM13] If G is a snark then scc ( G ) ≤ m + o ( m )For the class of cyclically 4-edge-connected graph this conjecture gives asignificant strengthening of Conjecture 1. In [EM13] a snark with scc ( G ) = m + 2 was constructed..Apart from the computational results of [BGHM13] we also know thatConjecture 1.2 holds for snark families where the girth grows with n , sinceJackson [Jac94] proved that scc ( G ) ≤ ( + g ) m for 2-edge-connected cubicgraphs of girth g and g ≥
8. Additional conditions for having short cyclecovers with different structural properties were given in [Ste12].In this paper we will show that Conjecture 1.2 is true for several classicalfamilies of snarks by relating short cycle covers to the perfect matching indexof a cubic graph, see Section 5, where we also show that graphs with lowperfect matching index satisfy Zhang’s conjecture. We will also prove thatfor graphs with long cycles, of length at least n − k , where k ≤
9, wehave scc ( G ) ≤ m + f ( k ), and that graphs of oddness 2 have short cyclecovers if they have small 2-factors. Most of our results a proven by showingthat the graphs at hand have cycle double covers which contain 2-regularsubgraphs which are close to a 2-factor in size. Finally we discuss some openproblems motivated by our current investigation, look at the connection toJaeger’s Petersen-colouring conjecture, and point out a perhaps unexpectedrigidity in the structure of shortest covers of some snarks and graphs of lowconnectivity. Our proofs will make use of the following simple observations, neither ofwhich are new and have been observed on different occasions by severalother authors.
Lemma 2.1.
If a cubic graph G has a 2-regular subgraph C and a CDC C such that C ⊂ C then G has a (1 , -cycle cover of length m − | C | Typically we will be interested in cycle covers with length close to theshortest possible value m and for the very shortest cases we can reversethe conclusion of the preceding lemma.2 emma 2.2. If G has a cycle cover F of length m + k , where k = 0 , then the edges of weight 1 form a 2-regular subgraph C of length n − k suchthat C ∪ F is a cycle double cover of G Proof.
Given a cycle covering F we define the weight of an edge e to be thenumber of cycles from F which contain e , and the weight w ( v ) of a vertex v to be the sum of the weights of the edges incident to v . The length of F is then given by (cid:80) v w ( v ).The lowest possible weight of a vertex is 4, which happens if the incidentedges have weights 1,1 and 2. We now note that in a covering of theselengths all edges must have weight 1 or 2, since (cid:80) v w ( v ) ≥ m + 2 if thereexists at least one edge of weight 3.Next we note that the edges of weight 1 must form a 2-regular subgraph C of G with length n − k , and C together with F form a cycle double coverof G In [God88] Goddyn posed what is now known as the strong cycle doublecover conjecture: Given a cycle C in a 2-connected cubic graph there isa cycle double cover of G which contains C . In [HM12, BGHM13] thisconjecture was verified for snarks on first 32 and then n ≤
36 vertices.In [HM12] this result was used in order to prove that if the circumference circ ( G ) of G , ie the length of the longest cycle in G , is close enough to n then G has a CDC. By following the method of that proof we can also provethat graph with large circumference have short cycle covers. Theorem 3.1.
There is a function f ( k ) such that if the strong cycle doublecover conjecture is true for cubic graphs with at most k vertices then for G such that circ ( G ) = n − k we have scc ( G ) ≤ m + f ( k ) , and in general f ( k ) ≤ k .Proof. Let C be a longest cycle of G and let G (cid:48) be the cubic graph whichis homeomorphic to the graph obtained by deleting all chords of C . If | C | = n − k then G (cid:48) can have at most 4 k vertices, which happens when thevertices not in C form an independent set, and is 2-connected. If the strongcycle double cover conjecture is true for cubic graphs on at most 4 k verticesthen G (cid:48) has a CDC C which contains the cycle corresponding to C .The cycle C together with all its chords is homeomorphic to a hamil-tonian cubic graph G (cid:48)(cid:48) with C giving the hamiltonian cycle. This graphhas a 3-edge-colouring with colour 1 and 2 on the edges on the hamiltoniancycle and colour 3 on the chords. Each pair of colours define a 2-regular3raph and those three graphs form a CDC C of G (cid:48)(cid:48) which contains the cyclecorresponding to C .Now ( C ∪ C ) \ C forms a cycle double cover of G which contains the2-regular graph given by the colours 1 and 3 in G (cid:48)(cid:48) . This graph has size atleast n − k and the theorem follows by Lemma 2.1 Corollary 3.2. f ( k ) exists for k ≤ , and f (0) = 0 , f (1) = 1 Proof. f [0) = 0 since in this case the graph is hamiltonian. By Theorem Bof [FH09] a cyclically 4-edge-connected cubic graph with an n − C has a CDC which contains C , so f (1) ≤ f (1) ≥
1. In [BGHM13] it was verified that the strongcycle double cover conjecture holds for n ≤
36, so f ( k ) exists for k ≤ Recall that the oddness o ( G ) of a cubic graph G is the minimum numberof odd cycles in any 2-factor of G . Graphs with oddness 0 are 3-edge-colourable and graphs with oddness 2 and 4 are known to have cycle doublecovers [HK95, HM05, Huc01] and also behave well with respect to othercycle cover/decomposition properties [Mar12].For some graphs of oddness 2 we can show an optimal bound on thelength of the shortest cycle cover. Theorem 4.1. If G is cyclically 4-edge-connected and has a 2-factor withexactly two components then scc ( g ) ≤ m + 2 .If there are three consecutive vertices on one the two cycles in the 2-factorwith neighbours on the other cycle then scc ( g ) ≤ m + 1 . There are graphs of this type with scc ( G ) = m + 1, e.g the Petersengraph, so the bound in the theorem is at most 1 away from being sharp. Thesecond case of the theorem will for example apply if one of the two cycles isan induced cycle in G , as in the Petersen graph. Proof.
Let C , C be a 2-factor with exactly two components and let e i , i =1 , , C and C . Let G (cid:48) be the cubicgraph which is homeomorphic to the graph given by E ( G ) \ { e , e , e } .Now G (cid:48) has a 2-factor consisting of two even cycles D and D , corre-sponding to C and C , and so has a 3-edge-colouring with colours 1 and 2on the edges of D and D , and colour 3 on the perfect matching M givenby E ( G ) \ E ( D ∪ D ). We may assume that at least two of the three edgeson which the endpoints of e , e , e were in C have colour 1, if necessary byswitching the colour 1 and 2 on D , and likewise on D .The colouring now gives a cycle double covering of H (cid:48) , by the cycles givenby each pair of the three colours, and a corresponding cycle double covering4 of E ( G ) \ { e , e , e } such that C ⊂ C and C ⊂ C (cid:48) . Similarly C , C together with e , e , e gives a graph with a CDC C such that C ⊂ C and C ⊂ C . Now C ∪ C \ { C , C } is a CDC of G which includes thecycles given by the colours 1 , G (cid:48) , and by our choice of colouring thelength of this 2-regular subgraph is at least n −
2, so by Lemma 2.1 we have scc ( G ) ≤ m + 2If the second case of the theorem applies then we can choose the colouringof the cycle in the condition such that all three of those edge endpoints lieon an edge of colour 1, thereby missing at most one vertex on the othercycle.It is possible to generalize the preceding proof to the situation where the2-factor has some even components as well, but the bound becomes weaker. Theorem 4.2. If G is cyclically 4-edge-connected and has a 2-factor F withexactly two odd components C and C , and the total number of edges in thethree shortest disjoint paths from C to C in the multigraph obtained bycontracting each cycle in F to a vertex is d , then scc ( G ) ≤ m + 2 d .Proof. Let C and C be the two odd cycles in F . By Menger’s theorem wecan find three disjoint paths P , P , P from C to C in G . We can nowproceed as in the previous proof but using the three paths instead of theedges e i . Since some cycles in F can intersect more than one of the threepaths P i we cannot necessarily use the colouring modification used in theprevious proof and so each edge of the three paths may contribute 2 edgeswhich are not included in the (1,3)-coloured subgraph, thus adding 2 d tothe length of the covering.It is possible to make the bound for h ( d ) somewhat stronger but sincethe method does not seem likely to give a sharp result, and quickly becomeslengthy, we have not included that analysis here. Definition 5.1.
The perfect matching index τ ( G ) of a bridgeless cubicgraph is the smallest integer k such that there exists perfect matchings M , . . . , M k such that ∪ k M k = E ( G )Note that G is 3-edge-colourable if and only if τ ( G ) = 3. If Fulkerson’sconjecture [Ful71] is true then τ ( G ) ≤ G .The perfect matching index was studied in more detail in [FV09], where itsvalue for several families of snarks were determined, and in [BGHM13] it wasshown that all but two of the snarks with n ≤
36 vertices have τ ( G ) = 4. Proposition 5.2.
Let M , M be two perfect matchings in a cubic graph G .Then M ∆ M induces a subgraph H of G which consists of disjoint evencycles. roof. Since every vertex of H either has degree 0 or it has degree 2 andone edge belongs to M and the other to M the result follows. Theorem 5.3.
Let G be a cubic graph. Then G has a 5-CDC where onecolour class is a 2-factor if and only if τ ( G ) ≤ .Proof. If τ ( G ) = 3 then G is 3-edge-colourable and the CDC given by thethree pairs of edge colour classes is a 3-CDC in which each colour pair definea 2-factor.If τ ( G ) = 4 then we let M = { M , M , M , M } be a set of four perfectmatchings such that (cid:83) i =1 M i = E ( G ). By the pigeonhole principle, eachvertex is incident to exactly one edge which is covered twice by the perfectmatchings in M . Therefore the set of edges covered by exactly two perfectmatchings is a perfect matching which we denote by M . Now consider A = { M ∆ M , M ∆ M , M ∆ M , M ∆ M } . By Proposition 5.2 A is a set ofeven subgraphs. If e ∈ M then e is covered twice by A and if e (cid:54)∈ M thenit is covered exactly once by A . Hence A ∪ { E ( G ) \ M } is a 5-CDC of G inwhich one colour class is a 2-factor.Conversely assume that G has a 5-CDC C = { C , C , C , C , F } where F is a 2-factor of G . Then M i = ( F ∩ C i ) ∪ (( E ( G ) \ F ) \ C i ) is a perfectmatching for i = 1 , . . . ,
4. Let M = { M , M , M , M } . If e ∈ F then eis covered by exactly one perfect matching in M and if e ∈ E ( G ) \ F itis covered by exactly two elements of M . Hence M is a perfect matchingcover of G with four perfect matchings.In combination with our earlier results this shows that graphs with τ ( G ) ≤ Theorem 5.4. If τ ( G ) ≤ then scc ( g ) = m Proof.
By Theorem 5 G has a CDC which contains a 2-factor and then thetheorem follows by Lemma 2.1.Theorem 5.4 was implicitly present already in the PhD Thesis of Celmins[Cel85], with a different proof and terminology, but as far as we have beenable to determine he did not publish it. That a low perfect matching indeximplies the existence of a 5-CDC, and thereby a short cycle cover, wasrecently also shown by Steffen in [Ste12].By this theorem the snarks in several of the classical snark families haveshort cycle covers. We refer to e.g [FV09] for the full definition of thesesnarks families. Corollary 5.5.
The Flower snarks, Goldberg Snarks, and Permutationsnarks which are not the Petersen graph have scc ( G ) = m roof. In [FV09] it was proven the snarks in each of these families have τ ( G ) = 4.As a corollary we also get a partial result on the conjecture of Zhang[Zha97] mentioned in the introduction, as already pointed out by Steffen[Ste12]. Corollary 5.6. If τ ( G ) ≤ then the shortest cycle cover of G is a (1 , -cover. Finally we note that the situation for cubic graphs of cyclic edge-connectivityexactly 2, or 3, is not as simple as one might hope. As before we only need toconsider graphs which cannot be three-edge-coloured, so the only differencefrom the snark case is the cyclic connectivity.There are standard reductions for cutting a cubic graph into two smallergraphs G and G at a 2-edge or 3-edge cut, adding in extra edges and/ora vertices to make the component graphs cubic. Ideally one could hopethat a shortest covering of G could be built from shortest coverings of G and G . However this is not always the case. In order for this compositionconstruction to work one would need to know that for any edge e in G or G there is a shortest cover where e has a specified weight, in order to makesure that the cycle covers agree when we compose the two smaller graphs toform G . This is true for 3-edge-colourable cubic graphs, and the Petersengraph, but it is not true for one of the two snarks on 18 vertices which hasan edge with only one possible weight in all shortest coverings. Observation 5.7.
In every 2-factor of the snark S , shown in the left partof Figure 5.1, which can be extended to a cycle double covering there is acycle which includes the edge ( a, b ) . Hence the edge ( a, b ) has weight 1 inevery shortest cycle cover of S . The observation can be proven by either a case analysis or a direct com-puter search.Using this observation we can easily construct graphs where each com-ponent coming from a 2-edge cut has shortest cycle cover of length m , whilethe original graph does not. Example 5.8.
Delete the edge ( a, b ) from three copies of the snark S andthen join them up to four additional vertices as shown on the right in Figure5.1, where the dark circles are the copies of S , to form a new graph S .If we cut away the bottom copy of S , and add the edge ( a, b ) to thiscopy and the edge ( u, v ), from this graph we are left with one large cubicgraph S which has a 2-factor which corresponds to a 2-factor in each of thecopies of S which includes the deleted edge ( a, b ), and so has a cycle cover of7 redag 24 maj 2013 u v Figure 1:length | E ( S ) | . However each such cycle cover gives weight 2 to the edge( u, v ), which corresponds to the 2-edge cut used to separate the third copyof S , and so cannot be composed with a shortest cycle cover of S . Hence scc ( S ) ≥ | E ( S ) | + 1In [FV09] it was proven that the perfect matching index of a graph G witha 2-edge cut is always at least the maximum of the perfect matching indexof the two graphs obtained by cutting G at a 2-edge cut. From Theorem 5.4and Example 5.8 it follows that the perfect matching index of a graph canbe strictly greater than that of components coming from a 2-edge cut. A we have seen snarks with scc ( G ) larger than m are rare among the smallsnarks, and the classical snark families, however we expect the minimum toincrease for larger snarks. Conjecture 6.1.
There exist snarks with n vertices such that scc ( G ) ≥ m + log n , for infinitely many n . If we restrict our attention to snarks with bounded oddness it seemsreasonable that the cover length should stay close to the minimum possible,as for the snarks of high circumference.
Conjecture 6.2.
There is a constant c such that scc ( G ) ≤ m + c if o ( G ) =2 . Problem 6.3.
Is there are function h such that scc ( G ) ≤ m + h ( o ( G )) ?
8n [Jae80] Jaeger made conjecture with far reaching consequences for thestructure of cubic graphs. A
Petersen -colouring of a cubic graph G is a map c : E ( G ) → E ( P ), where P is the Petersen graph, such that if e , e , e areincident edges in G then c ( e ) , c ( e ) , c ( e ) are incident edges in P . Jaegerconjectured that all 2-connected cubic graphs have a Petersen-colouring.This is trivially true for 3-edge-colourable graphs and in [BGHM13] theconjecture was verified for all snarks on n ≤
36 vertices.Given a Petersen-coloring c of G and a cycle cover F of P we can con-struct a cycle cover of G by taking the inverse image under c of each cyclein the cover. Each edge in P makes a contribution to the length of c − ( F )which is equal to its weight in F times the number of edges mapped ontoit by c . Using this observation, and saying that c is balanced if the samenumber of edges in G is mapped onto every edge in P , we have: Theorem 6.4.
If a cubic graph G has a P -colouring then scc ( G ) ≤ m . Ifthe colouring is not balanced then scc ( G ) ≤ m − P to which more than the average numberedges is mapped and use that as the cycle which is covered only once in theshortest cover of P . Corollary 6.5.
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