Structure of Graphs and Number of Maximum Matchings

1. Structure of Graphs and Number of Maximum Matchings Liu Yan School of Mathematics & South China Normal University

2. Topics • Gallai-Edmonds Decomposition • Ear Decomposition • Cathedral Construction

3. Gallai-Edmonds Decomposition • Basic Concepts: Factor-critical graph G: G-u has a p.m. for any u V(G). Bipartitie graph G=(X,Y) with positive surplus(about X): |(S)|>|S| for any SX. Deficiency of G: the number of vertices uncovered by a maximum matching, denoted by def(G).

4. Gallai-Edmonds partition: Denote by D(G) the set of all vertices in G which are not covered by at least one maximum matching of G. Denote by A(G) the set of vertices in V(G)-D(G) adjacent to at least one vertex in D(G). C(G)=V(G)-A(G)-D(G).

5. Gallai-Edmonds Decomposition Theorem

6. For any maximum matching M of G, the number of non trivial components of <D(G)> which are M-near full is denoted by nc(M). Let nc(G)=nc= max{nc(M)} where the maximum is taken over all maximum matchings M. Theorem (Liu Yan and Liu Guizhen, 2002 ) For any graph G, υf(G)=1/2(|V(G)|–def(G)+nc) = 1/2(|V(G)|–c(D(G))+|A(G)|+nc)

7. Theorem (Liu Yan , submitted ) • Let B be a subset of E(G). If v(G-B)=v(G), then c(D(G-B))-|A(G-B)|=c(D(G))-|A(G)|; • D(G-B) is a subset of D(G), C(G) is a subset of C(G-B) • When G is bipartite, A(G-B) is a subset of A(G)

8. Ear Decomposition Let G′ be a subgraph of a graph G. An ear: 1) a path P of odd length; 2) end-vertices of P are in V(G′); 3) internal vertices of P are not in V(G′). An ear system: a set of vertex-disjoint ears. Ear-decomposition: G′ =G1G2G3 … Gr= G where each Gi is an ear system and Gi+1is obtained from Gi by an ear system so that Gi+1 is 1-extendable or factor-critical.

9. Theorem (Liu Yan and Hao Jianxiu, 2002) Let G be a 2-connected factor-critical graph. Then G has precisely |E(G)| maximum matchings if and only if there exists an ear decomposition of G starting with a nice odd cycle C: G=C+P_1+…… +P_k, satisfying that two ends of P_i are connected in G_{i-1} by a pending path of G with length 2, where G_0=C and G_i=C+P_1+… +P_i for 1≤i≤ k.

11. Theorem (Liu Yan , 2004 ) Let G be a factor-critical graph with c blocks. Then (1) m(G)=|V(G)| if and only if every block of G is an odd cycle.  (2) m(G)=|V(G)|+1 if and only if all blocks of G are odd cycles but one, say such block H, and H is a theta graph θ(l1, l2, l3) with l1=2 and the path of length l1 is a pending path of G.

12. Theorem (Liu Yan , 2006 ) • Let G be a 2-connected factor-critical graph.Then m(G)=|V(G)|+2 if and only if G are the following.

13. Cathedral Construction Elementary graph: its allowed lines form a connected graph. Saturated graph G: G+e has more perfect matchings than G for all lines eE(G). Tutte’s Theorem: A graph G has a p.m. if and only if o(G-S) ≤ |S| for any S V(G) Barrier set S: a vertex-set S satisfying o(G-S) = |S|

15. Results about elementary graphs • If G is elementary, the set of maximal barriers is a partition of V(G). • If G is saturated elementary and S is a maximal barrier, G has |S| maximal barriers which are singleton. • If G is saturated elementary, the subgraph induced by a maximal barrier is complete.

16. Cathedral construction of saturated graphs

17. Theorem (Xiao Lan and Liu yan, 2006) 1.Let G be an elementary graph with 2n vertices and having exactly two perfect matchings, then |E (G) |2n+n(n–1)/2. 2. If f(n) denotes the maximum number of edges in a graph on 2n vertices having exactly two perfect matchings, then f(n)=n2+1.

18. Conjecture (Liu, 2006) If f(n) denotes the maximum number of edges in a graph on 2n vertices having exactly three perfect matchings, then f(n)=n2+2.

19. Thank you!!