Even cycles and even 2-factors in the line graph of a simple graph

Let G be a connected graph with an even number of edges. We show that if the subgraph of G induced by the vertices of odd degree has a perfect matching, then the line graph of G has a 2-factor whose connected components are cycles of even length (an even 2-factor). For a cubic graph G, we also give a necessary and sufficient condition so that the corresponding line graph L(G) has an even cycle decomposition of index 3, i.e., the edge-set of L(G) can be partitioned into three 2-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index m in 2d-regular graphs is also addressed.


Introduction
The graphs considered in this paper are simple and finite. We refer to [3] for graph theory notation and terminology which are not introduced explicitly here. In particular, we shall use the following terminology: a cycle is a connected 2-regular graph; the number of edges in a cycle is called its length; a cycle is even if it has even length. The vertex-disjoint union of cycles, that are contained in a graph G, is a 2-regular subgraph of G. A spanning 2-regular subgraph of G is a 2-factor of G. A 2-regular subgraph (in particular, a 2-factor) is even if its connected components are even cycles. A cycle decomposition of a graph G is a partition of the edge-set of G into cycles. It is known that a graph G possesses a cycle decomposition if and only if every vertex of G has even degree (see [20]).
Cycle decompositions satisfying additional conditions are widely studied (see for instance the survey paper [12]). In this paper we consider even cycle decompostions, i.e., cycle decompositions whose elements are even cycles. A necessary condition for the existence of an even cycle decomposition of a graph G is that every vertex has even degree and every block of G has an even number of edges. The existence of even cycle decompositions in planar graphs is completely settled. It is known that for planar graphs the necessary condition is also sufficient (see [17]). For a non-planar graph G a result by Zhang [24] holds, namely, if G satisfies the necessary condition and has no K 5 -minor, then G possesses an even cycle decomposition.
Given a graph G with a cycle decomposition, we can color the cycles of the decomposition in such a way that cycles sharing a vertex receive distinct colors. Each colored class is then a 2-regular subgraph of G. If m is the minimum number of colors that are required in such a coloring, then we say that the decomposition is a cycle decomposition of index m. Since each colored class is a 2-regular subgraph of G, a cycle decomposition of index m provides a partition of the edge-set of G into m 2-regular subgraphs, i.e., a 2-regular subgraph decomposition of G of cardinality m. If each colored class is a 2-factor of G, then a 2-regular subgraph decomposition of G is known as a 2-factorization of G. There might be more than one way to color the cycles of a decomposition by m colors, i.e., a cycle decomposition of index m might provide more than one 2-regular subgraph decomposition of cardinality m. Obviously, a cycle decomposition of index m consisting of c cycles provides a 2-regular subgraph decomposition of cardinality m ′ , for every m ≤ m ′ ≤ c. By Petersen's Theorem [16], a 2d-regular graph possesses a 2-factorization. Therefore, every 2d-regular graph has at least one cycle decomposition of index d. An arbitrary cycle decomposition of a 2d-regular graph has index m ≥ d. If a cycle decomposition of index m satisfies additional properties, then these properties may well be inherited by the corresponding 2-regular subgraph decomposition of cardinality m. In particular, an even cycle decomposition of index m provides an even 2-regular subgraph decomposition of cardinality m. We are interested in determining the minimum number m of even 2-regular subgraphs which partition the edge-set of a graph or, equivalently, the minimum value of m taken over all even cycle decompositions of index m. Our motivation for formulating this problem is explained in Section 1.1. By the previous remarks, the index of an even cycle decomposition of a 2d-regular graph satisfies the inequality m ≥ d. A class 1 regular graph of degree 2d possesses an even cycle decomposition of index d (since the edge-set of a class 1 regular graph of degree 2d can be partitioned into 2d perfect matchings, we can pair the matchings and find d even 2-factors). It is easy to show that the converse is also true. Hence, a 2d-regular graph has an even cycle decomposition of index d if and only if it is class 1. If the graph is class 2, then every even cycle decomposition has index m ≥ d + 1 (see [7] for the definition of graphs of class 1 and 2 according to Vizing's Theorem).
We restrict our attention to 4-regular graphs and in particular to 4regular line graphs. We study the existence/non-existence of even 2-factors and even cycle decompositions in these graphs. We recall that the vertices of the line graph L(G) of a graph G correspond to the edges of G. Two vertices of L(G) are adjacent if and only if the corresponding edges of G share a vertex. Hence, if G is 3-regular, then L(G) is 4-regular. The reason of our interest in 4-regular graphs arises from the study of a chromatic parameter (the palette index) and will be explained in Section 1.1. We consider line graphs because these graphs are completely characterized by a list of nine forbidden subgraphs (see [1]) which reduces to seven forbidden subgraphs if the graphs are regular of degree 4. We shall see that some properties holding for line graphs can be used to determine the structure of the original graphs and conversely (see for instance Proposition 2,3,5). Our results involve other decompositions, namely, even star decompositions and P 2 -decompositions. An even star decomposition D of a simple graph G is a partition of the edge-set of G into stars K 1,2h whose centre has even degree 2h ≥ 2 (even stars). We do not require that the stars in D are pairwise isomorphic. If each star in D is a path P 2 (a path with two edges), then we say that D is a P 2 -decomposition of G.
We briefly summerize the contents of this paper. Section 2.2 is devoted to the existence of an even 2-factor in the line graph of an arbitrary graph G. We note that the line graph of a connected graph G with an even number of edges has an even 2-factor if and only if the graph G has a pair of disjoint P 2 -decompositions ("disjoint" here is understood in a set-theoretical sense, i.e., the disjoint decompositions have no common member). We prove the following sufficient condition: if the subgraph K induced by the vertices of odd degree has a perfect matching, then G has a pair of disjoint P 2decompositions. In Section 2.1 we always assume that G is cubic and prove that the existence of an even cycle decomposition in L(G) is equivalent to the existence of three subgraphs of G satisfying certain conditions (see Proposition 5). Every class 1 cubic graph possesses such subgraphs. An almost straighforward argument shows that the line graph of a class 1 cubic graph has an even cycle decomposition of index 3 (see Corollary 2). By the results in [13], the line graph of a class 1 cubic graph with an even number of edges is a 4-regular graph of class 1. Hence, in this case, we can find an even cycle decomposition of smallest index, namely, of index 2 (see the previous remarks on graphs of class 1).
The existence of the three subgraphs for a class 2 cubic graph does not seem to be an immediate consequence of the definition: we namely give a sufficient condition for the existence of such subgraphs (see Proposition 6 and 7). We use these conditions to show that the line graphs of some class 2 cubic graphs (flower snarks, Blanuša snarks, Goldberg snarks and others) admit even cycle decompositions of index 3. The numerous examples of 4-regular line graphs with an even cycle decomposition of index 3 seem to confirm a conjecture by Markström [15] stating that a 4-regular graph on 2n + 1 vertices asymptotically almost surely decomposes into one cycle of length 2n and two further even cycles, i.e., it has an even cycle decomposition of index 3. We note that our results hold for graphs on an even number of vertices, as well.
Our constructions are described in terms of "even orientations" (more details in Section 2). In particular, in Proposition 1, we obtain a refinement of a classical result of Kotzig [14] stating that a connected graph has an even orientation if and only if it has an even number of edges.

A related question.
A (proper) edge-coloring of a graph G defines at each vertex v the set of colors of its incident edges, the so called palette of the vertex v. The minimum number of distinct palettes taken over all proper edge-colorings of G is the palette index of G and is denoted byš(G) (see [9]). As remarked in [9], the palette index of a regular graph is different from 2 and satisfies the inequalities 1 ≤š(G) ≤ χ ′ (G) where χ ′ (G) denotes the chromatic index. Moreover,š(G) = 1 if and only if the graph G is class 1. Consequently, the possible values for the palette index of a class 2 cubic graph are 3 and 4. By the results in [9], a class 2 cubic graph has palette index 3 if and only if it has a perfect matching, otherwise the palette index is 4. Hence, for every admissible value it is possible to find an example of a class 2 cubic graph with the required palette index.
What is the behavior of d-regular graphs with d > 3 in this respect? In other words, given an integer r, 3 ≤ r ≤ d + 1, is it possible to find a d-regular graph with palette index r? The case d = 4 is studied in [4] and it is shown that for every integer r, 3 ≤ r ≤ 5, it is possible to find a 4-regular graph with palette index r.
While studying the palette index of 4-regular graphs we observed the following phenomenon: a 4-regular graph of class 2 has palette index 3 if and only if it has an even 2-factor or an even cycle decomposition of index 3. We can exhibit instances of 4-regular graphs with an even 2-factor and/or an even cycle decomposition of index 3 (see [4]). We can also give examples of 4-regular graphs with palette index 4 and 5, but none of them admits an even cycle decomposition. We do not know examples of graphs, with palette index larger than 3, possessing an even cycle decomposition. If such a graph exists, then it has no even 2-factor and every even cycle decomposition has index m > 3 (otherwise the palette index should not exceed 3). The problem of finding a 4-regular graph all of whose even cycle decompositions have index larger than 3 was another motivation for this paper, as we found no result concerning the index of an even cycle decomposition in the literature.

Even orientations.
An orientation of a simple graph G = (V (G), E(G)) is a directed graph − → G = (V (G), D(G)) obtained from G by specifying, for each edge [u, v] of G, an order on its end-vertices. If a = (u, v) is an arc of − → G , then the vertex u is the tail of a and v is the head of a.

Theorem 2. [14, Theorem 3] If G is a connected graph with an even number of edges, then G has an even orientation.
Combining Theorem 1 and Theorem 2, one can see that a connected graph has an even orientation if and only if it has an even number of edges. This result is also known as follows: "a connected graph has a P 2 -decomposition if and only if it has an even number of edges". In the next section we analyse the relationship between even orientations and P 2decompositions in greater detail. We give some remarks that will be used to prove our results.

Even orientations and even decompositions.
Let − → G be an even orientation of a graph G. If v is a vertex of − → G with indegree 2h > 0, then the underlying edges of the incoming arcs in v form a subgraph of G isomorphic to the star K 1,2h (a path P 2 for h = 1). If we consider all the vertices of − → G with positive even indegree, then we obtain an even star decomposition of G. Obviously, also the converse is true: an even star decomposition of G gives rise to an even orientation of − → G (for every star . Therefore there exists a one-to-one correspondence between even orientations and even star decompositions. If G is a graph with an even number of edges and maximum degree 3, then the one-to-one correspondence is between even orientations and P 2 -decompositions of G (for graphs with maximum degree 3 a star K 1,2h is a path P 2 ). For an arbitrary graph G, with an even number of edges, if the star decomposition D corresponding to the even orientation of − → G contains at least one star D ∼ = K 1,2h with 2h > 2, then we can pair in an arbitrary way the edges of D and find more than one P 2 -decomposition of G arising from the same even orientation.
Let D 1 , D 2 be a pair of disjoint even star decompositions of a graph G.
We say that a pair of even orientations of the same graph G are disjoint if they satisfy the aforementioned property. A pair of disjoint even star decompositions gives rise to a pair of disjoint even orientations and conversely. We can now prove the following result. Proposition 1. Let G be a connected graph with an even number of edges. Let K be the subgraph of G induced by the vertices of odd degree. If K has a perfect matching M , then there exists a pair of disjoint even orientations of share only the arcs whose underlying edge belongs to M .
Proof. By Theorem 2, the graph G has an even orientation − → G 1 . We change the orientation of all arcs whose underlying edges are not in M . Parity is preserved.
Corollary 1. Let G be a connected graph with an even number of edges. Let K be the subgraph of G induced by the vertices of odd degree. If K has a perfect matching, then there exists a pair of disjoint even star decompositions of G and also a pair of disjoint P 2 -decompositions of G.
Proof. The existence of a pair of disjoint even star decompositions, say D 1 , D 2 , follows from Proposition 1, since a pair of disjoint even orientations of G corresponds to a pair of disjoint even star decompositions of G. If G is a graph with maximum degree ≤ 3, then an even-star decomposition is nothing but a P 2 -decomposition, hence the assertion follows in this case.
We consider G with maximum degree larger than 3 and prove that G has a pair of disjoint P 2 -decompositions. Let − → G 1 , − → G 2 be the disjoint even orientations of G arising from Proposition 1 and corresponding to the disjoint even star decompositions D 1 , D 2 , respectively. By Proposition 1, the arc- share only the arcs whose underlying edges belong to M . Hence, for every D ∈ D 1 and D ′ ∈ D 2 , the cardinality of is either empty or consists of a single edge in M ). Consequently, if D ∈ D 1 and D ′ ∈ D 2 are stars with the same centre v, then by pairing the edges in D and in D ′ we obtain a set A and a set A ′ containing exactly |E(D)|/2 and |E(D ′ )|/2 paths P 2 , respectively. The sets A and A ′ share no path P 2 , since the cardinality of E(D) ∩ E(D ′ ) is at most 1. Hence, D 1 and D 2 provide a pair of disjoint P 2 -decompositions of G.

Even 2-factors.
A P 2 -decomposition of a graph G corresponds to a perfect matching in the line graph L(G). More generally, a set of k paths P 2 in G, which are pairwise edge-disjoint, corresponds to a matching of cardinality k in L(G). Conversely: since an edge of L(G) corresponds to a pair of adjacent edges in G, a matching of cardinality k in L(G) (respectively, a perfect matching in L(G)) corresponds to a set of k paths P 2 in G which are pairwise edge-disjoint (respectively, to a P 2 -decomposition in G). We can prove the following results. Proof. Assume that L(G) has an even 2-factor F , then we can alternately color the edges of F and obtain two edge-disjoint perfect matchings, say M 1 and M 2 , of L(G). Each perfect matching M i , i = 1, 2, corresponds to a P 2 -decomposition of G, say D i . Since M 1 and M 2 are edge-disjoint, the corresponding P 2 -decompositions share no path P 2 , i.e., D 1 and D 2 are disjoint. It is easy to see that the converse is true as well.
Proposition 3. Let G be a connected graph with an even number of edges. Let K be the subgraph of G induced by the vertices of odd degree. If K has a perfect matching, then the line graph L(G) has an even 2-factor.
Proof. It follows from Corollary 1 and Proposition 2.
The following property is a straightforward consequence of Proposition 3.
Proposition 4. Let G be a connected cubic graph with an even number of edges. If G has a perfect matching, then the line graph L(G) has an even 2-factor.
The existence of a perfect matching, in a connected graph with an even number of edges and no vertex of even degree, is a sufficient condition for the existence of an even 2-factor in the corresponding line graph. The condition is not necessary: for instance, the graph G in Figure 1(a) is a cubic graph with 24 edges and no perfect matching whose line graph has an even 2factor. More specifically, the graph G in Figure 1(a) has 3 subgraphs, say N 1 , N 2 and N 3 , that contain the vertex u and are isomorphic to the graph N in Figure 1(b); the graph N has 8 edges and a perfect matching by Proposition 1, the graph N has a pair of disjoint P 2 -decompositions; therefore, each subgraph N i has a pair of disjoint P 2decompositions, say D i,1 and D i,2 with i = 1, 2, 3; the sets by Proposition 2, the line graph L(G) has an even 2-factor.
We can also give some examples of graphs with no perfect matching whose line graph has no even 2-factor. For instance, the graph G in Figure  2 is a cubic graph with 42 edges and no perfect matching whose line graph has no even 2-factor. More specifically, every even orientation of G contains the arcs (u, v), (w, v), i.e., every P 2 -decomposition of G contains the P 2path (u, v, w); hence G cannot have a pair of disjoint P 2 -decompositions; by Proposition 2, the line graph L(G) cannot have an even 2-factor. Figure 1: (a) A cubic graph with no perfect matching whose line graph has an even 2-factor; (b) the graph N .
u v w Figure 2: A cubic graph with no perfect matching whose line graph has no even 2-factor.
3 Even cycle decompositions for the line graph of a cubic graph.
The definition of disjoint even orientations of a cubic graph G can be read as follows: two even orientations of G, say − → By the one-to-one correspondence between even orientations and P 2decompositions we can write the following statement in terms of even orientations or P 2 -decompositions. Conversely: assume that L(G) has an even cycle decomposition D = {D 1 , D 2 , D 3 } of index 3. In each even 2-regular subgraph D i , i = 1, 2, 3, we alternately color the edges of each even cycle and obtain a pair of edgedisjoint matchings, say M i,1 , M i,2 , whose cardinality is |V (D i )|/2. The edges in M i,j , with j = 1, 2, yield a set D i,j containing exactly |V (D i )|/2 paths P 2 of G that are pairwise edge-disjoint. The paths in D i,j form a subgraph H i,j of G admitting D i,j as a P 2 -decomposition. Since the matchings M i,1 , M i,2 have the same vertex-set, the subgraphs H i,1 and H i,2 of G coincide and we set   Proof. By Theorem 2, the graph H has an even orientation − → H . If the vertices u 1 and u n have the same indegree in − → H , then the assertion follows. Consider u 1 and u n with different indegree in − → H , i.e., u 1 has indegree 2 whereas u n has indegree 0 (or vice versa). Let S be the connected component of K containing the vertex u n (it may well happen that S = K). The vertices u 0 and u 1 belong to S. Since S is 2-connected, the edge [u 1 , u n ] does not belong to E(G). Moreover, there exist two internally disjoint paths of S connecting the vertices u n and v 0 , i.e., the vertices u n and v 0 belong to a cycle C 1 of S. We can always assume that C 1 does not contain the vertex u 1 , i.e., C 1 does not contain the edge [u 0 , u 1 ] (if C 1 contains the edge [u 0 , u 1 ] we can take the chord [u 0 , v 0 ] and find a cycle of S containing the vertices u n , v 0 , but not the vertex u 1 ). We use C 1 to construct a new even orientation − → H 0 of H: starting from − → H , we leave unchanged the direction on the arcs in − → H whose underlying edges do not belong to C 1 ; we reverse the direction on the arcs in − → H with underlying edge belonging to C 1 . The new orientation − → H 0 is even: the vertices not in C 1 do not change their set of incoming arcs; the vertices in C 1 change the direction on exactly two of their arcs, hence the number of incoming arcs is always even. Moreover, u n and u 1 have the same indegree in − → H 0 and the assertion follows. Proof. We prove that Proposition 5 is verified. We set H 1 = H and denote by H 2 the graph obtained from G by deleting the vertex u 0 . We denote by − → H 1,1 ) whose underlying edges belong to M ′ 2 ; we reverse the direction on the arcs in D( − → H 1,1 ) whose underlying edges belong to ( Since we are assuming that u 1 and u n have indegree 2 in − → H 1,1 , we add the arcs (u i , u i+1 ), with 1 ≤ i ≤ n − 1 (see Figure 5). One can easily verify that − → H 2,1 is an even orientation of H 2 . Since M ′ 1 is a perfect matching on the subgraph of H 2 induced by the vertices of degree 3 and 1, we can apply Proposition 1 and find an even orientation − → H 2,2 of H 2 which is disjoint from − → H 2,1 (see Figure 5). By the same proposition, the arc-sets share only the arcs whose underlying edges belong to M ′ 1 . We show that condition (iii) of Proposition 5 is verified. The orientations − → H 1,r and − → H 3,s , with r, s = 1, 2 are disjoint on every share only the edges that are incident to u 0 (see Figure  3 and 4). From Figure 3    As an application of Proposition 6, consider the following example. Example 1. The Petersen graph, denoted by GP (5, 2), is a class 2 cubic graph with 15 edges. We denote by the edge-set of GP (5, 2) (the subscripts are taken modulo 5). We denote by H the subgraph of GP (5, 2) obtained by deleting the edges [u i , u i+1 ], with 1 ≤ i ≤ 3, of the cycle C = (u 0 , u 1 , . . . , u 4 , u 0 ). The subgraph K 1 of H has a perfect matching Figure 7); the subgraph K 2 of H is 2-connected and has a perfect matching Figure 4: The arcs in the even orientations of H 1 possessing at least one vertex in V (C). Note that the edge [u 1 , u n ] does not belong to the graph G, since the connected component of H 1 containing u n is 2-connected. The vertex v 0 is adjacent to at most one of the vertices u 1 , u n , since each connected component of H 1 has an even number of edges. For the same reason, the cycle C contains no chord. The arcs whose underlying edges belong to M 1 (respectively, to M 2 ) are depicted with a bold line (respectively, with a dashed line). v 0 u 0    Figure 7). Following the proof of Proposition 6, we can construct the even orientations − → H i,r with i = 1, 2, 3 and r = 1, 2 in Figure 7. Hence the line graph of GP (5, 2) has an even cycle decomposition of index 3. As an application of Proposition 7, consider the flower snark J 5 .

see the bold edges in
Example 2. The flower snark J 5 is a class 2 cubic graph with 30 edges. We denote by V = {x i , y i , v i , u i : 0 ≤ i ≤ 4} the vertex-set of J 5 . The edge-set of J 5 can be defined as follows: the vertices x i , y i , v i , u i induce a star K 1,3 with centre v i ; the vertices u 0 , u 2 , . . . , u 4 induce a cycle (u 0 , u 1 , . . . , u 4 , u 0 ) of length 5; the vertices x i , y i , with 0 ≤ i ≤ 4, induce a cycle (x 0 ,x 1 ,. . . , x 4 , y 0 , y 1 ,. . . , y 4 , x 0 ) of length 10.
We consider the subgraph of J 5 consisting of the cycle C = (u 0 , u 1 , . . ., u 4 , u 0 ) with the pendant edge [u 0 , v 0 ]. Following the proof of Proposition 7, we denote by H 1 the subgraph of J 5 obtained by deleting the edges Figure 9: A pair of disjoint even orientations of the subgraph H 2 in J 5 . situation in some known classes of such graphs. In the previous section we have already seen three examples for which the answer is affirmative. To our knowledge, the earliest class 2 cubic graphs that have been discovered are: the Petersen graph [16], the first and second Blanuša snark [2], the Descartes snark [6] and the Szekeres snark [18]. In [10] Isaacs found new class 2 cubic graphs: the BDS class, the flower snarks and the double star. The snarks of Blanuša, Descartes and Szekeres inspired the construction of the BDS class and are contained in it. The BDS class also contains other snarks that were constructed by other authors. See for instance the Celmins-Swart snarks [5] and the Watkins snark [21]. Watkins also generlized the construction of the two Blanuša snarks [22]. Further known families of class 2 cubic graphs are the two families of snarks found by Loupekine and described by Isaacs in [11] and the Goldberg snarks [8].
The sufficient conditions in Proposition 6 and 7 are satisfied for the following graphs: the double star, the generalized Blanuša snarks, the flower snarks, the Goldberg snarks and for the snarks of Szekeres, Descartes, Celmins-Swart, Watkins, Loupekine LP 0 [19]. Hence, the line graphs of these snarks have an even cycle decomposition of index 3. We checked the same sufficient conditions on some class 2 cubic graphs on 12, 14, 28 and 30, respectively (they are usually considered to be "trivial" snarks since their girth is < 5). Some of them do not verify these sufficient conditions, but verify the necessary and sufficient condition in Proposition 5. Therefore, also in this case we can find an even cycle decomposition of index 3 in the corresponding line graph. We give a detailed proof of the existence of an even cycle decomposition of index 3 for the generalized Blanuša snarks and for the Goldberg snarks. For the flower snarks, one can generalize the construction in Example 2. For the remaining cases we prefer to omit the proof.
The generalized Blanuša snarks are divided into two families: the generalized Blanuša snarks of type 1 and those of type 2. A generalized Blanuša snark of type 1, denoted by B 1 n , is constructed as follows: consider n − 1 copies B 1 , B 2 , . . . , B n−1 of the block B in Figure 11 and exactly one copy of the graph A 1 ; label the vertices a, a In the construction of a Blanuša snark of type 2, denoted by B 2 n , the block A 1 is replaced by the block A 2 . The generalized Blanuša snarks B 1 2 and B 2 2 are known as the first Blanuša snark and the second Blanuša snark, respectively. A generalized Blanuša snark has 3(4n + 1) edges. Proof. The proof is based on Proposition 6. The block B in Figure 11 has a cycle C = (a, y, x, b, z, a) of length 5 with a pendant edge. We delete the edges [a, y], [x, y], [x, b] in the block B 1 of each generalized Blanuša snark and obtain a subgraph H of B j n , j = 1, 2. The subgraphs K 1 and K 2 of H have a perfect matching M 1 and M 2 , respectively, that can be defined as in Figure 12 (bold edges belong to M 1 , dashed edges belong to M 2 ) and are edge-disjoint. The graph H is connected, the subgraph K 2 is 2-connected. We can apply Proposition 6 and find an even cycle decomposition of index 3 in the corresponding line graph.
x y z Figure 11: The basic blocks in the construction of the Blanuša snarks. A Goldberg snark G n is a cubic graph with 12n edges and can be defined as follows: consider and odd integer n ≥ 5 and n copies B 1 , B 2 , . . . , B n , of the block B in Figure 13(a); label the vertices a, b, c, d, x, y, z, t of each copy B i by a i , b i , c i , d i , x i , y i , z i , t i , respectively; add the edges [a i , a i+1 ], [c i , b i+1 ], [y i , x i+1 ] with 1 ≤ i ≤ n (the subscripts are taken modulo n). We have the following property. Proof. We show that Proposition 7 holds. The block B in Figure 13(a) has a subgraph given by a cycle C = (x, y, z, d, t, x) of length 5  of H have a perfect matching M 1 and M 2 , respectively, that are defined as follows: for n ≥ 5, color the edges of the blocks B 1 , B 2 , B n−2 , B n−1 , B n as in Figure 13(c); for n > 5 and for every odd index i, 3 ≤ i ≤ n − 4, color the edges of the blocks B i , B i+1 as in Figure 13

Final remarks.
The existence of a 2-connected 4-regular graph possessing only even cycle decompositions of index larger than 3 remains an open problem. Futhermore, in view of the results obtained for 4-regular line graphs, we can add another related problem, namely, the existence of a 2-connected cubic graph that does not verify the necessary and sufficient condition in Proposition 5. The line graph of such a graph would admit only even cycle decompositions of index larger than 3 or no even cycle decomposition at all. As remarked in Section 1, our results seem to confirm a conjecture in [15] stating that a 4-regular graph on an odd number of vertices asymptotically almost surely has an even cycle decomposition of index 3. Moreover, our results hold for graphs on an even number of vertices as well. In [15], it is also conjectured that the edge-set of the line graph of a 2-connected cubic graph decomposes into cycles of even length. Finding a 2-connected cubic graph not fulfilling Proposition 5 appears thus to be a hard problem.
As remarked in Section 1, the existence of an even cycle decomposition of index 3 in a 4-regular graph of class 2 is a particular case of the more general problem about the existence of an even cycle decomposition of minimum index m in a 2d-regular graph of class 2 for which the inequality m ≥ d + 1 holds. We note that the necessary and sufficient condition in Proposition 5 can be generalized to (2d+1)-regular graphs by taking 3d ≥ 3 subgraphs H i , 1 ≤ i ≤ 3d, verifying the three conditions in Proposition 5. This generalization provides an even cycle decomposition of index 3d for the corresponding 4d-regular line graph. Therefore, the minimum value m for the index of an even cycle decomposition of a 4d-regular line graph satisfies the inequalities 2d + 1 ≤ m ≤ 3d. For 4-regular graphs of class 2, the minimum value m for the index of an even cycle decomposition is 3 and Proposition 5 describes how to find 4-regular graphs possessing an even cycle decomposition whose index is as small as possible. For 2d-regular graphs of class 2, d > 2, a generalization of Proposition 5 (which would be quite straightforward) would not yield a description of how to find 2d-regular graphs possessing an even cycle decomposition whose index is as small as possible. It would only give an upper bound for the minimum value m in the case of 4d-regular line graphs.