Structure of Graphs and Number of Maximum Matchings

Liu Yan

School of Mathematics

&

South China Normal University

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Topics

Gallai-Edmonds Decomposition Ear Decomposition Cathedral Construction

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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 SX.

Deficiency of G: the number of vertices uncovered by a maximum matching, denoted by def(G).

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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).

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Gallai-Edmonds Decomposition Theorem

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)

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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)

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Ear DecompositionLet 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+1 is obtained from Gi by an ear

system so that Gi+1 is 1-extendable or factor-critical.

G’

P1 P2

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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.

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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.

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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.

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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 eE(G).

Tutte’s Theorem: A graph G has a p.m. if and only if o(G-S) ≤ |S| for any SV(G)Barrier set S: a vertex-set S satisfying o(G-S) = |S|

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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.

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Cathedral construction of saturated graphs

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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.

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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.

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