Maximum matchings in regular graphs
arXiv:1308.2269v2 [math.CO] 8 Nov 2013
Dong Ye
∗
Abstract It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan [4, 5] that: Every graph G with ∆(G) − δ(G) ≤ 1 has a maximum matching M such that any two M -unsaturated vertices do not share a neighbor. Early results obtained in [4, 6, 7] leave the conjecture unknown only for k-regular graphs with 4 ≤ k ≤ 6. All counterexamples for k-regular graphs (k ≥ 7) given in [6] have multiple edges. In this paper, we consider the conjecture for both simple k-regular graphs and k-regular graphs with multiple edges. We show that the conjecture holds for all simple k-regular graphs, and for k-regular graphs (k = 4 or 5) with multiple edges.
1
Introduction
Graphs considered in this paper may have multiple edges, but no loops. A matching M of a graph G is a set of independent edges. A vertex is covered by a matching M if it is incident with an edge of M . A vertex is M -saturated if it is covered by M , and M -unsaturated otherwise. A matching M is said to be maximum if for any other matching M ′ , |M | ≥ |M ′ |. A matching M is perfect if it covers all vertices of G. If G has a perfect matching, the every maximum matching is a perfect matching. The maximum and minimum degrees of a graph G are denoted by ∆(G) and δ(G), respectively. Mkrtchyan, Petrosyan and Vardanyan [4, 5] made the following conjecture. Conjecture 1.1 (Mkrtchyan et. al. [4, 5]). Let G be a graph with ∆(G)−δ(G) ≤ 1. Then G contains a maximum matching M such that any two M -unsaturated vertices do not share a neighbor. This conjecture is verified for subcubic graphs (i.e. ∆(G) = 3) by Mkrtchyan, Petrosyan and Vardanyan [4]. However, Picouleau [7] find a counterexample to the conjecture, which is a simple bipartite graph with δ(G) = 4 and ∆(G) = 5. Recently, Petrosyan[6] constructs examples to show that the conjecture does not hold for graphs G with δ(G) = k and ∆(G) = k + 1 for k = 3 or k ≥ 5, and for all k-regular graphs when k ≥ 7. So the conjecture remains unkown only for k-regular graphs for 4 ≤ k ≤ 6. There is no affirmative result known for the conjecture except for subcubic graphs [4]. Note that, the counterexample for almost regular graph given by Picouleau [7] is simple. But all counterexamples for k-regular graphs for k ≥ 7 obtained by Petrosyan [6] have multiple edges. In ∗ Department
of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132; Email:
[email protected]
1
this paper, we consider the conjecture for both simple k-regular graphs and k-regular graphs with multiple edges. First we show that Conjecture 1.1 does hold for all simple k-regular graphs. Theorem 1.2. Let G be a simple k-regular graph. Then G has a maximum matching M such that any two M -unsaturated vertices do not share a neighbor. Further, we show that Conjecture 1.1 is true for general 4-regular graphs and 5-regular graphs which may have multiple edges. Theorem 1.3. Let G be a k-regular graph for k = 4 or 5. Then G has a maximum matching M such that any two M -unsaturated vertices do not share a neighbor. Unfortunately, we are unable to prove the conjecture for 6-regular graphs with multiple edges, which is the only case remaining unknown.
2
Preliminaries
Let G be a graph and v be a vertex of G. The neighborhood of v is set of all vertices adjacent to v, denoted by N (v). The degree of v is dG (v) = |N (v)|. If there is no confusion, we use d(v) instead. For X ⊆ V (G), let δ(X) := min{d(v)|v ∈ X} and ∆(X) := max{d(v)|v ∈ X}. The neighborhood of X is defined as N (X) := {y|y is a neighbor of a vertex x ∈ X}. Let M be a matching of G. A vertex subset X is M -saturated if M covers all vertices of X. For two subgraphs G1 and G2 of G, the symmetric difference of G1 ⊕ G2 is defined as a subgraph with vertex set V (G1 ) ∪ V (G2 ) and edge set (E(G1 ) ∪ E(G2 ))\(E(G1 ) ∩ E(G2 )). Use [G1 , G2 ] to denote the set of all edges joining vertices of G1 to vertices of G2 . A matching of G is called a near-perfect matching if it covers all vertices except one. If a graph G has a near perfect matching, then G has odd number of vertices. A graph is factor-critical if for any vertex v of G, G − v has a perfect matching. Clearly, a factor-critical graph has a near-perfect matching. Let D be the set of all vertices of a graph G which are not covered by at least one maximum matching, and A, the set of all vertices in V (G) − D adjacent to at least one vertex in D. Denote C = V (G) − A − D. The graph induced by all vertices in D (resp. A and C) is denoted by G[D] (resp. G[A] and G[C]). The following theorem characterized the structures of maximum matchings of graphs, which is due to Gallai [2] and Edmonds [1]. Theorem 2.1 (Gallai-Edmonds Structure Theorem, Theorem 3.2.1 in [3] on page 94). Let G be a graph, and A, D and C are defined as above. Then: (1) the components of the subgraph induced by D are factor-critical; (2) the subgraph induced by C has a perfect matching; (3) if M is a maximum matching of G, it contains a near-perfect matching of each component of G[D], a perfect matching of G[C] and matches all vertices of A with vertices in distinct components of G[D].
2
Contract every component of G[D] to a vertex and let B be the set of all these vertices. Let G(A, B) be the bipartite graph obtained from G with bipartition A and B. Then, a maximum matching of G(A, B) is corresponding to a maximum matching of G, and vice versa. Before processing to prove our main results, we needs some results for maximum matchings of bipartite graphs G(A, B). Theorem 2.2 (Hall’s Theorem, Theorem 1.13 in [3] on page 5). Let G(A, B) be a bipartite graph. If for any S ⊆ A, |N (S)| ≥ |S|, then G has a matching M covering all vertices of A. The following techincal lemmas are needed in proof of our main results. Lemma 2.3. Let G(A, B) be a bipartite graph such that every maximum matching of G(A, B) cover all vertices of A. Let W ⊆ B such that δ(W ) ≥ ∆(A). Then G(A, B) has a maximum matching M covering all vertices W . Proof. Let M be a maximum matching of G(A, B) such that the number of vertices of W covered by M is maximum. If M covers all vertices of W , we are done. So assume that there exists an M -unsaturated vertex x ∈ W . Note that, for any U ⊆ W , |N (U )| ≥ |U | because d(w) ≥ δ(W ) ≥ ∆(A) ≥ d(v) for any w ∈ U and v ∈ N (U ) ⊆ A. By applying Hall’s Theorem on the subgraph induced by W and N (W ), it follows that G has a matching M ′ covering all vertices of W . Let M ⊕ M ′ be the symmetric difference of M and M ′ . Every component of M ⊕ M ′ is either a path or a cycle. Since x is not covered by M , it follows that x is an end vertex of some pathcomponent P of M ⊕ M ′ . Let y be another end vertex of P . Note that every vertex of A is covered by an edge of M and every vertex of W is covered by an edge of M ′ . So y ∈ B\W . Then let M ′′ = M ⊕ P . Then M ′′ is a maximum matching of G which covers x and all vertices covered by M except y. Note that y ∈ B\W and x ∈ W . Hence M ′′ covers more vertices of W than M , a contradiction to the maximality of the number of vertices of W covered by M . This completes the proof. Let Pk be a path of order k (i.e., k vertices). A {P2 , P3 }-packing S of G is a subgraph of G such that every component of S is either P2 or P3 . A matching of G is a {P2 , P3 }-packing without P3 -components. Lemma 2.3 can be generalized for {P2 , P3 }-packings as follows. Lemma 2.4. Let G(A, B) be a bipartite graph and W ⊆ B. If every maximum matching of G covers all vertices of A and ∆(A) ≤ 2δ(W ), then G(A, B) has a {P2 , P3 }-packing S such that S covers all vertices of A and W , and dS (v) ≤ 1 for any v ∈ B. Proof. Suppose to the contrary that G has no such a {P2 , P3 }-packing. Since every maximum matching of G covers all vertices of A, G has a {P2 , P3 }-packing S such that: (1) dS (v) ≤ 1 for any v ∈ B; and (2) S covers all vertices of A. Choose a {P2 , P3 }-packing S of G satisfying (1), (2) and (3) Subject to (1) and (2), S covers the maximum number of vertices of W . 3
Let x ∈ W be a vertex that is not covered by S. Then for any y ∈ N (x), the degree of y in S satisfies dS (y) = 2. Otherwise, let yx′ be a component of S. Then replacing yx′ by xyx′ in S will increase the number of vertices covered by S, a contradiction to (3). Now orient all edges of S from A to B and all other edges from B to A. Let Gx be the maximal subgraph of G consisting of all vertices v such that there is a directed path from x to v in G. Since Gx is maximal, there is no directed edge from Gx to Gcx := G − Gx . For any vertex v ∈ Gx ∩ B, a directed path joining x to v passes through an edge of S incident with v. Note that, dS (v) = 1 for any v ∈ B. It follows that S does not contain directed edges from Gcx ∩ A to Gx ∩ B. Further, there is no edge from Gcx ∩ A to Gx ∩ B since all edges not in S are oriented from B to A. So the only edges between Gx and Gcx are directed from Gcx ∩ B to Gx ∩ A. Hence N (V (Gx ) ∩ B) = V (Gx ) ∩ A. By (3), every vertex y of V (Gx ) ∩ A has degree two in S, and every vertex z of V (Gx ) ∩ B is in W . Otherwise, S ⊕ P is a {P2 , P3 }-packing of G containing more vertices from W where P is a directed path from x to y (or z), a contradiction to (3). So we have 2|V (Gx ) ∩ A| = |V (Gx ) ∩ B| − 1, and δ(Gx ∩ B) ≥ δ(W ). By N (V (Gx ) ∩ B) = V (Gx ) ∩ A, it follows that δ(W )(|V (Gx ) ∩ B|) ≤ ∆(A)|V (Gx ) ∩ A|
k by |[Qi , A]| < k. Hence every component of Qi for i ≥ t contains a good vertex because of |[Qi , A]| < k. Let X = {vi |vi ∈ Qi for i ≥ t and N (vi ) ⊆ V (Qi )}. Contract all components Qi into a vertex qi and let H be the bipartite graph with bipartition A and B = {qi |i = 1, 2, ...}. Let W := {qi |dH (qi ) ≥ k}, the set of vertices corresponding to such Qi 6
with |[Qi , A]| ≥ k. By Lemma 2.3, H has a maximum matching M covering all vertices of W and all vertices of A. Note that each component Qi is factor-critical. For each component Qi (i < t), it has a near-perfect matching Mi that does not cover a vertex which is covered by M . And for each component Qi (i ≥ t), it has a near-perfect matching Mi which does not cover a good vertex. By Gallai-Edmonds Structure Theorem, let M ′ be a maximum matching of G such that M ′ matches all vertices of A with vertices in distinct components Qi ’s as the same as M , and contains a perfect matching of G[C] and the near-perfect matching Mi of each component of Qi . Then any two M ′ -unsaturated vertices belong to X and hence do not share a common neighbor since Qi ∩ Qj = ∅ for i 6= j. This completes the proof of the theorem. Now we are going to prove Theorem 1.3. Proof of Theorem 1.3. All graphs discussed here may contain parallel edges. Prove the theorem by dealing two different cases k = 4 and k = 5. Case 1. k = 4. For i ≥ t, |[Qi , A]| < k = 4. Since |[Qi , A]| ≡ k = 4 (mod 2), it follows that |[Qi , A]| = 2 for i ≥ t. Hence Qi for i ≥ t is not a singleton. Further, Qi for i ≥ t has at least three vertices because Qi is factor-critical. So every component Qi for i ≥ t has a good vertex vi . Let X = {vi |vi ∈ Qi with i ≥ t and N (vi ) ⊆ V (Qi )}. Contract all components Qi into a vertex qi and let H be the bipartite graph with bipartition A and B = {qi |i = 1, 2, ...}. Let W := {qi |dH (qi ) ≥ 4}. By Lemma 2.3, H has a maximum matching M covering all vertices of W . By Gallai-Edmonds Structure Theorem, G has a maximum matching M ′ such that M ′ ∩[A, D] = M , and M ′ contains a perfect matching of G[C] and a near perfect matching of every Qi that does not cover either a good vertex or a vertex covered by M . So all M ′ -unsaturated vertices are good vertices and hence belong to X. Hence any two M ′ -unsaturated vertices do not share a neighbor since Qi ∩ Qj = ∅ for i 6= j. This completes the proof of the theorem for k = 4. Case 2. k = 5. Since G is 5-regular and each Qi has odd number of vertices, it follows that |[Qi , A]| ≡ 1 (mod 2). For i ≥ t, |[Qi , A]| < 5 and hence Qi is not a singleton. Note that if |[Qi , A]| = 1, then Qi contains a good vertex. A component Qi with |[Qi , A]| = 3 may also contain a good vertex. Order Qi ’s for i ≥ t such that, for some integer β ≥ t, every Qi with i ≥ β contains a good vertex, and Qj for t ≤ j < β does not contain a good vertex. It follows that |Qj | = 3 = |[Qi , A]| for t ≤ j < β, and hence every vertex of Qj (t ≤ j < β) has a unique neighbor in A. Contract all components Qi into a vertex qi and let H be the bipartite graph with bipartition A and B = {qi |i = 1, 2, ...}. Let W := {qi |dH (qi ) ≥ 5} (i.e., i < t), and U := {qi |dH (qi ) = 3}. Note that G is 5-regular. So ∆(A) ≤ 5. By Gallai-Edmonds Structure Theorem, every maximum matching of H is A-saturated. By Lemma 2.5, H has a {P2 , P3 }-packing S such that: (1) S covers all vertices of A, W and U ; (2) dS (v) ≤ 1 for any v ∈ B; 7
(3) S has no component containing two vertices of W . Decompose the {P2 , P3 }-packing S to two matchings M and M ′ as follows: for each P3 -component Q, choose an edge e of Q that is not incident with a vertex of W , and put e into M ′ and another edge of Q into M ; and put all P2 -components into M . By the properites (1), (2) and (3) of S, M and M ′ satisfy the following properties: • M covers all vertices of A and W ; • M ′ covers all M -unsaturated vertices in U . Every component Qi with |[Qi , A]| ≥ 5 (i.e., i < t) is corresponding to a vertex of W and hence contains a vertex vi that is incident with an edge in M . Every component Qj with |[Qj , A]| = 3 is corresponding to a vertex of U and hence contains a vertex that is either incident with an edge in M or incident with an edge in M ′ . All component Qp with |[Qp , A]| = 1 contains a good vertex. In the following, we are going to construct a desired maximum matching M ∗ of G: any two M ∗ -unsaturated vertices do not share a common neighbor. Let X be a set of vertices choosen by the following process: (i) choose a good vertex from each component Qi (i ≥ β) that has no vertex incident with the edges in M ; (ii) choose a vertex vj from each Qj (t ≤ j < β) such that vj is incident with an edge of M ′ . By Gallai-Edmonds Structure Theorem, G[C] has a perfect matching M0 , and each component Qi has a near perfect matching Mi that does not cover the vertex that is either incident by M or contained in X. So let c(D) [ Mi ). M ∗ := M ∪ ( i=0
∗
By Gallai-Edmonds Structure Theorem, M is a maximum matching of G. By the construction of M ∗ , X is the set of all M ∗ -unsaturated vertices. Let x1 , x2 ∈ X. If one of them is a good vertex, then they do not share a neighbor. So suppose that both of them are not good vertices. Then the components Qim containing xm (m = 1, 2) have no good vertices by (i) in the construction of X (i.e., t ≤ im < β). So |Qim | = 3 = |[Qim , A]| and every vertex of Qim (m = 1, 2) has exactly one neighbor in A. Since x1 and x2 are incident with two edges in the matching M ′ , their neighbors in A are distinct. So x1 and x2 do not share a neighbor. This completes the proof of the theorem for 5-regular graphs. Remark. The conjecture for 6-regular graphs with multiple edges seems to require new ideas or a substantial refinement of Lemma 2.5.
References [1] J. Edmonds, Paths, trees and flowers, Canad. J. Math 17 (1965) 449-467. [2] T. Gallai, Maximale systeme unabh¨ angiger kanten, Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ozl. 9 (1964) 401-413. [3] L. Lov´asz and M.D. Plummer, Matching Theory, North Holland, Amsterdan, 1986.
8
[4] V.V. Mkrtchyan, S.S. Petrosyan and G.N. Vardanyan, On disjoint matchings in cubic graphs, Discrete Math. 310 (2010) 1588-1613. [5] V.V. Mkrtchyan, S.S. Petrosyan and G.N. Vardanyan, Corrigendum to “On disjoint matchings in cubic graphs” [Discrete Math. 310 (2010) 1588-1613], Discrete Math. 313 (2013) 2381. [6] P.A. Petrosyan, On maximum matchings in almost regular graphs, preprint, arXiv:1202.0681. [7] C. Picouleau, A note on a conjecture on maximum matching in almost regular graphs, Discrete Math. 310 (2010) 3646-3647.
9
