Jian Cheng, Cun-Quan Zhang & Bao- Xuan Zhu

Even factors of graphs Jian Cheng, Cun-Quan Zhang & Bao- Xuan Zhu Journal of Combinatorial Optimization ISSN 1382-6905 DOI 10.1007/s10878-016-0038-4 1 23

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DOI 10.1007/s10878-016-0038-4 Even factors of graphs Jian Cheng 1 Cun-Quan Zhang 1 Bao-Xuan Zhu 2 Springer Science+Business Media New York 2016 Abstract An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Favaron and Kouider (J Gr Theory 77:58 67, 2014) showed that if a simple graph G has an even factor, then it has an even factor F with E(F) 7 16 ( E(G) +1). This ratio was improved to 4 7 recently by Chen and Fan (J Comb Theory Ser B 119:237 244, 2016), which is the best possible. In this paper, we take the set of vertices of degree 2 (say V 2 (G)) into consideration and further strengthen this lower bound. Our main result is to show that for any simple graph G having an even factor, G has an even factor F with E(F) 4 7 ( E(G) +1) + 1 7 V 2(G). Keywords Spanning subgraph Maximum even factor 2-Factor Extremal graph theory 1 Introduction Graphs in this paper are finite and simple. Let G be a graph with the vertex set V (G) and the edge set E(G). For each vertex v V (G), define E G (v) to be the set of Partially supported by an NSF grant DMS-1264800, NSA grants H98230-12-1-0233 and H98230-14-1-0154, and National Natural Science Foundation of China grant #11571150. B Bao-Xuan Zhu bxzhu@jsnu.edu.cn Jian Cheng jiancheng@math.wvu.edu Cun-Quan Zhang cqzhang@math.wvu.edu 1 Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA 2 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, P China

edges incident with v in G.Thedegree d G (v) of v in V (G) is E G (v).theminimum degree of G is denoted by δ(g). We denote V k (G) (resp. V k (G)) the set of vertices of degree k (resp. at least k) ing. For any two subgraphs A and B of G, the union A B is the subgraph of G induced by E(A) E(B), and the symmetric difference, denoted by A B, is the subgraph of G induced by (E(A) E(B)) \ (E(A) E(B)). For any X V (G), G[X] is the subgraph of G induced by X. Aneven factor of G is a spanning subgraph of G in which each vertex has a positive even degree. If an even factor is 2-regular, then we call it a 2-factor of G. For convenience, a l-circuit (resp. ( l)-circuit) is a circuit whose length is l (resp.atleastl). Petersen s theorem (1891) says that every bridgeless cubic graph G has a 2-factor F with E(F) = 2 E(G) 3. This classical theorem was extended to all non-cubic graphs by Fleischner (1992). Theorem 1.1 (Fleischner 1992) If G is a bridgeless graph with minimum degree δ(g) 3, then G has an even factor. By applying the Matching Polytope Theorem of Edmonds (1965) and the (vertex) Splitting Lemma (Fleischner 1976), the size of a maximum even factor was first estimated by Lai and Chen (1999). Theorem 1.2 (Lai and Chen 1999) If G is a bridgeless graph with minimum degree δ(g) 3, then G has an even factor F with E(F) 2 3 E(G). elaxing the requirements of bridgeless and minimum degree, Favaron and Kouider (2014) showed the following improvement. Theorem 1.3 (Favaron and Kouider 2014) If a graph G has an even factor, then it has an even factor F with E(F) 9 16 ( E(G) +1). Chen and Fan (2016) recently improved this ratio to 4 7, which is the best possible. The extremal graphs can be obtained from trees by blowing up each vertex with pairwise disjoint K 4 (complete graph of order 4). Theorem 1.4 (Chen and Fan 2016) If a graph G has an even factor, then it has an even factor F with E(F) 4 7 ( E(G) +1). Our interest in this problem was motivated by the relation between V 2 (G) (the set of vertices of degree two) and even factors. Let F be any maximum even factor of a graph G and subdivide one edge of F. Note that F generates an even factor F of

the resulting graph G and the numbers of edges of F and G both increase by 1. The inequality in Theorem 1.4 still holds, but the gap between the left side and right side is larger with the increase in V 2 (G). Based on above observations, we conjecture that when V 2 (G) increases, the size of a maximum even factor gets larger. In 2016, Chen and Fan introduced an optimized edge-coloring technique. In this paper, we adopt this technique and further introduce the concept of Q-subgraphs. Combined with the discharging method, we confirm aforementioned conjecture and strengthen Theorem 1.4 as follows. Theorem 1.5 If a graph G has an even factor, then it has an even factor F with E(F) 4 7 ( E(G) +1) + 1 7 V 2(G). 2 Proof of the main theorem Without loss of generality, all graphs considered in the proof are connected, since the inequality being considered is obviously additive over connected components. Proof of Theorem 1.5 Let (G, F) be a contra pair to Theorem 1.5 such that (1) E(G) is minimized; (2) subject to (1), m(g) = v V 3 (G) (d G(v) 3) is minimized; (3) subject to (1) and (2), E(F) is maximized; (4) subject to (1)-(3), the number of components of F is as small as possible. Claim 1 V 2 (G) is an independent subset of V (G). Proof Suppose that there are two vertices u,v V 2 (G) such that uv E(G). Let G (and F ) be the graph obtained from G (and F) by identifying u and v into a new vertex w and deleting the resulting loop. Clearly, E(G ) < E(G),w V 2 (F ), and F is an even factor of G. By the choice of (G, F), G has an even factor F such that E(F ) 4 7 ( E(G ) +1) + 1 7 V 2(G ). Let F be the subgraph of G induced by ( E(F ) E F (w) ) E G (u) E G (v). Since w V 2 (F ), F is an even factor of G with E(F ) = E(F ) +1. Note that E(G) = E(G ) +1 and V 2 (G) = V 2 (G ) +1. Hence, we have E(F ) ( 4 7 ( E(G ) +1) + 1 ) 7 V 2(G ) + 1 = 4 7 E(G) +1 7 ( V 2(G) 1) + 1 = 4 7 ( E(G) +1) + 1 7 V 2(G) + 2 7 > 4 7 ( E(G) +1) + 1 7 V 2(G) This contradicts the choice of (G, F).

Claim 2 Fisa2-factor of G. Proof Suppose that one component I of F contains a vertex v such that d F (v) = 2t 4. Since each component of F is an Valerian subgraph of G, we have an Valerian tour T from I. We split v into t vertices of degree 2, each of which is incident with two consecutive edges in T. Denote the resulting graphs by F (from F) and G (from G), respectively. By assumption (2), (G, F ) is not a contra pair since E(G ) = E(G) and m(g )<m(g). Thus G has an even factor F such that E(F ) 4 7 ( E(G ) +1) + 1 7 V 2(G ). Note that V 2 (G ) = V 2 (G) +t > V 2 (G).LetF be the subgraph of G induced by E(F ), then F is an even factor of G with E(F ) = E(F ). Furthermore, we have E(F ) 4 7 ( E(G ) +1) + 1 7 V 2(G ) = 4 7 ( E(G) +1) + 1 7 ( V 2(G) +t) > 4 7 ( E(G) +1) + 1 7 V 2(G) This contradicts the choice of (G, F). 2.1 { B, Y, }-edge-coloring of G In this part, we follow an edge-coloring introduced by Chen and Fan (2016) and define an {B, Y, }-edge-coloring ( ) of G as follows. For a maximum matching B of F, lety = E(F) \ B and = E(G) \ E(F). For simplicity, we use, B, and Y to denote the subgraphs induced by E(), E(B), and E(Y ), respectively. Note that F is a union of vertex-disjoint circuits by Claim 2. We will pick a particular maximum matching B out of F by choosing, for each odd circuit C of F, the vertex v not incident with any member of E(C) B according to the following rules. See Fig. 1 for an illustration. (a) if C is an odd circuit of F and F has more than one component, then let v be avertexofc such that d D (v) is as large as possible in D = G[V (C)] and v is incident with both Y -edges in C; (b) subject to (a), d (v) is as large as possible. Denote the collection of such vertices by V 0 (G). An edge e is called an B-edge ( Y -edge, -edge, respectively) if e B(e Y, e, respectively). Claim 3 B is a spanning subgraph of G. Proof Suppose that G has a vertex v V (G) V ( B). Then v is incident with only Y -edges and therefore v V 0 (G). By Claim 2, d G (v) = 2 and v is incident with precisely two Y -edges. By the definition of V 0 (G), there is an odd component C of F such that v V (C). By rule (b) in ( ), each vertex of C has degree 2 in G. This implies G = C, a contradiction to the choice of (G, F).

Fig. 1 The {B, Y, }-edge-coloring around an odd circuit of F,andthe selection of the vertex v Y v Y B B Y B Y Claim 4 Each circuit in B is alternately {, B}-edge-colored, and any two circuits are vertex-disjoint. Proof Suppose that Z is a circuit of B that is not alternately {, B}-edge-colored. Since B is a matching of F, Z F is an even factor larger than F, a contradiction to assumption (3). Suppose Z 1 and Z 2 are two non-disjoint circuits in B and denote K = Z 1 Z 2. Then K contains a vertex v of degree at least 3. Note that any vertex of degree 4 in K must be incident with two -edges and two B-edges contradicting that B is a matching. Thus d K (v) = 3 and each vertex of degree 3 is incident with one B-edge and two -edges (one is in Z 1 and one is in Z 2 ). This implies that Z 1 Z 2 contains a circuit of B in which two -edges are adjacent to each other. This contradicts that each circuit in B is alternately {, B}-edge-colored. Claim 5 If Z is a circuit of length 2μ of B, then the number of components of F intersecting with Z is at most max{μ 1, 1}. Proof Suppose that there is a circuit Z of B of length 2μ intersecting with μ(μ 2) components of F. By Claim 4, the circuits in B are alternately {, B}- edge-colored and pairwise vertex-disjoint. Thus F Z is an even factor of G. Since the (2μ)-circuit Z intersects with a set X of μ distinct components (circuits) of F, E(Z) E(C) =1 for every C X and therefore ( C X C) Z is a single circuit. Hence, F Z has (μ 1) components less than F, a contradiction to assumption (4). 2.2 Q-subgraphs in B In Claims 3, 4, and 5, the structure of circuits contained in G[ B] has been studied. Note that not every B-edge is contained in some circuit of G[ B]. Denote the collection of such B-edges by B 0 and call each of its elements an B 0 -edge. By Claim 3, each vertex of degree 2 must be incident with an B 0 -edge. By Claim 1, none of these B 0 -edges is incident with two vertices of degree 2. Thus we have B 0 V 2. Let Z ={Z 1,...,Z t } be the collection of all circuits in B.Lett k be the number of k-circuits of Z.LetT be the graph obtained from B by contracting each Z i into

T v r z 1 z 2 z 1 z 2 = T 1 T 2 T 1 T 2 Fig. 2 The forest T 1 T 2 = ( B)/Z and its extension T (a rooted tree) a vertex. By Claim 4, the circuits in Z are pairwise vertex-disjoint and therefore T is acyclic. Let {T 1,...,T a } be all components of T and assign an orientation to each T i as a rooted tree with the root z i (z i is selected arbitrarily). Then we further extend the rooted forest T 1 T a to be a rooted tree T as follows: add a new vertex v r as the root of T and add a new arc vr z i (z i is the root of T i ) for each 1 i a. See Fig. 2 for an illustration. Denote ={v r z i : 1 i a}. Finally, we construct a tree T from T by further contracting all remaining B- edges in T. Note that all those B-edges are not contained in any circuit of B and therefore are B 0 -edges. In T, each edge is an ( )-edge and each vertex except v r is incident with a unique incoming arc. Now we can classify the vertex set V (T ) {v r } as follows. Avertexx is a Type I vertex if it is contracted from a circuit of Z in B. Avertexx is a Type II vertex if it is contracted from an B 0 -edge in T. Avertexx is a Type III vertex if it is an original vertex in G. In this case, x is not incident with any B-edge. Thus we have x V 0. For any vertex x V (T ) {v r },let yx be the unique arc towards x in T, and let yu be the edge of G corresponding to yx in T.In B, we define the corresponding Q-subgraphs as follows. See Fig. 3 for an illustration. If x is Type I, let Z Z be the circuit contracted into x in T. The subgraph induced by the circuit Z and the edge yu is called a Type I Q-subgraph. (A Type I Q-subgraph looks like a tadpole with the body Z and the trail yu.) If x is Type II, let uv be the B 0 -edge contracted into x in T. The subgraph induced by the edges uv and yu is called a Type II Q-subgraph. (A Type II Q-subgraph is a path of length 3, not looking like a tadpole at all. However, for the sake of convenience, we still call the B 0 -edge uv the body and the edge yu the tail.) If x is Type III, then x = u remains as a vertex of B. The subgraph induced by the edge yx is called a Type III -subgraph. (A Type III Q-subgraph is a single edge yx. Similarly, we call the single vertex x the body and the edge yx the tail.) Denote the body and the tail of each Q-subgraph Q i by b(q i ) and t(q i ), respectively. Note that the collection of all Q-subgraphs, denoted by Q, is an edge partition of B.

y y y or u or u B 0 or x = u B Z B v B Type I. Type II. Type III. Fig. 3 Types I, II, and III Q-subgraphs Let Q I (Q II, Q III, respectively) be the collection of all Type I (Type II, Type III, respectively) Q-subgraphs. Then Q = Q I Q II Q III, Q II = B 0, and Q III = V 0 (G). 2.3 Charge and discharge In B, assign each vertex v of G with initial charge h V (v) = 1 and v r with initial charge h V (v r ) = 0; assign each edge e with initial charge h E (e) = 1 and each edge e B with initial charge h E (e ) = 0, respectively. Then we have equations as follows. Claim 6 + = n = V (G) = h V (v) = h V (Q i ) (1) v V (G) e h E (e) = Q i Q h E (Q i ) (2) Q i Q where h V (Q i ) = v b(q i ) h V (v) and h E (Q i ) = e E(Q i ) h E(e) for each Q i Q. Proof The proof of Eq. (2) is trivial. The Eq. (1) follows from Claim 3 that B is a spanning graph of G. Now we define a ratio to each Q-subgraph Q i as follows: r(q i ) = h V (Q i ) h V (Q i ) + h E (Q i ). Then we have the following observations. For each Q i Q I, it follows from Claim 4 that the body b(q i ) is an alternately {, B}-edge-colored circuit Z of Z. If Z =2μ, then r(q i ) = 2μ 2μ+(μ+1) with μ 2. This implies r(q i ) 4 7 since it is an increasing function on μ.

For each Q i Q II, the body b(q i ) is a single B 0 -edge. Thus r(q i ) = 2 3 (> 4 7 ). For each Q i Q III, the body b(q i ) is a single vertex. Thus r(q i ) = 1 2 (< 4 7 ). To increase the ratio of Q i Q III to 4 7, we either take some charge from a Type I Q-subgraph whose body is a ( 6)-circuit of Z, or take some charge from a Type II Q-subgraph. The following result guarantees that each Q i Q III can take some charge from an Q-subgraph in Q I Q II. Claim 7 Every odd circuit C of F either contains at least one B 0 -edge, or intersects with some ( 6)-circuit Z i of B. Proof Suppose that there is an odd circuit C of F such that each B-edge of C is contained in a 4-circuit of B. By Claim 5, each of those 4-circuits intersects with C only, no other components of F. Denote G[V (C)] =K. By rule (a) in ( ),wehave δ(k ) 3. Note that V (K ) = E(C) is an odd integer, so E(K ) > 2 3 V (K ) = 3 2 E(C). By Theorem 1.2, K has an even factor D such that E(D) 2 3 E(K ) > E(C). Thus D (F \ C) is an even factor of G larger than F, a contradiction. The Discharging ule Let C be an odd circuit of F with x V (C) V 0 (G). We redistribute the charge as follows. (1) If C intersects with some ( 6)-circuit Z of Z, then x takes 1 3 from some v V (C) V (Z). (2) If not, then by Claim 7, C contains at least one B 0 -edge uv. Letx take 1 3 from u. Now we have the following two observations. Claim 8 Both Eqs. (1) and (2) remain the same after redistributing the charge. Claim 9 The ratio of any Type III Q-subgraph is 4 7 now. For each Q i Q I, the body b(q i ) is a circuit Z of B. We define l Qi to be the length of Z, and k Qi to be the number of times of being discharged of Q i. Then we have the following result. Claim 10 For each Q i Q I, h V (Q i ) = 4 7 (h V (Q i ) + h E (Q i )) + 1 ( lqi k Qi 4) (3) 7 Moreover, r(q i ) 4 7 if k Q i l Qi 4. Proof Since Q i is discharged k Qi times, we have h V (Q i ) = l Qi 1 3 k Q i and h V (Q i ) + h E (Q i ) = 3 2 l Q i 1 3 k Q i + 1. Then Eq. (3) follows and r(q i ) 4 7 if k Q i l Qi 4.

Note that each B 0 -edge is contained in at most one odd circuit of F. Thus it can be discharged at most once. For each Q i Q II, we have the following result. Claim 11 h V (Q i ) = 4 h V (Q i ) + h E (Q i ) + 2 7 7 B 0\B 0 +1 7 B 0, Q i Q II Q i Q II Q i Q II (4) where B 0 is the set of B 0-edges with one endvertex being discharged once. Proof For each Q i Q II with b(q i ) B 0,wehave h V (Q i ) = 2 1 3 = 5 3 and h V (Q i ) + h E (Q i ) = 3 1 3 = 8 3. Hence, we have h V (Q i ) = 4 7 (h V (Q i ) + h E (Q i )) + 1 7. Similarly, for each Q i Q II with b(q i ) B 0 \ B 0,wehave h V (Q i ) = 4 7 (h V (Q i ) + h E (Q i )) + 2 7. Then Eq. (4) follows from above two equations. 2.4 The final step Assume the Discharging ule (1) and (2) occur x 1 and x 2 times, respectively. Thus we have B 0 x 2. By Claim 8, both Eqs. (1) and (2) remain the same. Combined with Claims 9, 10, and 11, wehave n = h V (Q) + h V (Q) + h V (Q) Q Q I Q Q II Q Q III = 4 h V (Q) + h E (Q) + 1 (l Q k Q 4) + 2 7 7 7 ( B 0 x 2 )+ 1 7 x 2 Q Q Q Q Q Q I = 4 ( n + + ) + 1 l Q k Q 4 + 1 7 7 7 (2 B 0 x 2 ) Q Q I Q Q I Q Q I (5) Note that l Q = Q Q I 2μt 2μ, μ 2 Q Q I k Q = x 1, and 1 = t 2μ. (6) Q Q I μ 2

Since n = E(F) and E(G) = E(F) +, itfollowsfromeqs.(5) and (6) that E(F) = 4 7 E(G) +4 7 + 1 4)t 2μ x 1 + 7 μ 2(2μ 1 7 (2 B 0 x 2 ). (7) By Claim 5 and the Discharging ule, for each Q Q I,wehavek Q l Q 2 1if l Q 6 and k Q = 0 otherwise. It follows from Eq. (6) that x 1 = ( ) lq k Q 2 1 = (μ 1)t 2μ 4)t 2μ Q Q I Q Q I : l Q 6 μ 3 μ 3(2μ That is, μ 2(2μ 4)t 2μ x 1 = μ 3(2μ 4)t 2μ x 1 0. It follows from Eq. (7) that E(F) 4 7 E(G) +4 7 + 1 7 (2 B 0 x 2 ). Since B 0 V 2 (G) and B 0 x 2,wehave (2 B 0 x 2 ) B 0 V 2 (G). Note that 1. Thus E(F) 4 7 ( E(G) +1) + 1 7 V 2(G). (8) This last inequality contradicts the choice of (G, F). 3emarks The family of extremal graphs in Theorem 1.4 shows that the first part of the lower bound in Eq. (8) (i.e. 4 7 ( E(G) +1)) cannot be improved any more. We do not know whether the coefficient 1/7of V 2 (G) is best possible or not, but the example of Fig. 4 shows that it cannot be larger than 2/7. Note that G 0 has a maximum even factor F 0 consisting of 4-circuits and satisfying E(F 0 ) = 4 7 ( E(G 0) +1) + 2 7 V 2(G 0 ). (9) Based on Eq. (9), we propose a conjecture as follows.

Fig. 4 Graph G 0 Conjecture 3.1 If a graph G has an even factor, then it has an even factor F with E(F) 4 7 ( E(G) +1) + 2 7 V 2(G). eferences Chen F, Fan G (2016) Maximum even factors of graphs. J Comb Theory Ser B 119:237 244 Edmonds J (1965) Maximum matching and a polyhedron with 0, 1-vertices. J es Natl Bur Stand Sect B 69:125 130 Favaron O, Kouider M (2014) Even factors of large size. J Gr Theory 77:58 67 Fleischner H (1976) Eine gemeinsame Basis für die Theorie der eulerschen Graphen und den Satz von Petersen. Monatsh Math 81:267 278 Fleischner H (1992) Spanning Eulerian subgraph, the splitting lemma and Petersen s theorem. Discrete Math 101:3 37 Lai H-J, Chen Z-H (1999) Even subgraphs of a graph, combinatorics, graph theory and algorithms. New Issues Press, Kalamazoo Petersen J (1891) Die Theorie der regulären graphs. Acta Math 15:193 220