GraPhs considered in this thesis are finite, undirected and ede. ROughly
spallng, the prOPerties we are concerned in this theSis are mainly as follow:
1. The degree sum conditions Of induced matching extendabe graPhs.
2. Mtwal induced matching tmextendable graPhs.
The orgedation of the thess is as follow'
In the first chaPter, we list some notatiOnS and introduce some toPics
about matching theory and the induced rnatching extendable gmphs. In the
second chaPter, the degree suxn conditions of IM-extendable graPhs, IM-ex-
tendable claw-free graphs and IM-extendab1e bipatite graPhs are presented. In
the third chapter, the mbomal induced matching unextendable graphs are
DrW Sum Conditions of 1nduced Matcbing Extendabe Graphs
MgE(G) is a matching of G, if v(e) n v(f) = o for every two dis-
tinct edges e, f e M. A matching is said to be the mchum .matching if it
contains the mbomum number of edges. If a matching covers all the vertices of
G, we call the matching a perfect matching of G. A matching M of G is in-
duced, if E (V(M) ) = M. A graph G is induced matching extendabe (short-
ly, IM-extendable), if every induced matching of G is included in a peffect
matching of G. Graph G is nextendable if G has a matching of sise n and ev-
ery such matching extends to (i. e. is a subset of ) a pedect matching in G. A
graPh G is bicritical if G -- u -- v has a perfect matching for evmp chOce of two
ponts u, v in G. Since PltnnIner[ 14 ] raised the problem of n-extendablity,
-- 1 --
znany resultS in the Wct M[5' 13' 14, 15 ]. Cameron gave the conCopt of
induced matching in . Ane ftmdamental resultS on induced matChing can
be found in [2, 4, 6]. In 1998, YUan deW the indUced tnaching ex-
tendable graphs, shothe, IM-extendable graPhs, and charaCterized them Part-
ly. Plummer has H the following reSul which invOlves degree stnns of
SetS of t indePendent poins, for t3.
LeUUna 1 - 1 SUPpee G is a k-connected graP with p points where p is
even and n is any inteqer sW 1n p/2. SuPpe further that there
W a t, 1tk -- 2n + 2 such that fOr all indePendent setS I = 1 wl, '..'
w,l having l II = t, it follow that .gld(wi)t((p --2)/2+ n) + 1. Then
(a) n = 1, G is bicritical (and hence 1 -- extendabl) and if
(b) n2, G is n-extendable.
The following result is a coroary of LerIuna 1. 1 (namely, the case when
t = 2).
Dere1lary 1 .2 Let G be a graph with p ponts, p even, and let n be an
integer, 1n p/2. SuPPOSe that for all pairS of nonadacent pontS u and v
in G, d(u) + d(v)7p +2n -- 1. Then if
(a) n == 1, G is bicritical (and hence l -- extendabe) and if
(b) n2, G is n-extendable.
Motivated by corOllary 1. 2, we obtain the degree sum conditions of in-
duced matching extendable graphs.
Tbeorem 1.3 Let G be a graph with 2n vertices. If for each pair of
nonadjacent vertices u and v in G, we have d(u ) + d (v)2rty1 -- 1, then
G is IMextendable. And this is best poSSble.
Theorem 1.4 SuPpoSe that G is a claw-free gr8Ph with 2n vertices. If
fOr each pair of nonadjacent vertices u and v in G, d (u ) + d (v)2n + 3,
then G is IM-extendable' And this is best mpible.
-- 2 --
Theorem 1 .5 Clawfree mforal planar graPh is IMextendable.
In the clawfree conditiOnS, we characterize all the graPhs with twUIn
degree sum 2n + 2 and 2n + 1 which are IMunextendable.
Theorem 1. 6 Let G be a biPartie graPh with biPhotion (A, B ),
where IA I = lBI = n - If for each pair Of nonadacent venices u and v in G,
we have d (u ) + d (v )rop1, then G is IM~extendabe.
Maxhoal 1nduced Matctw U--dable Grahs
A graph G is med IM-tmextendable if G is no induced matChing ex-
tendabe and, for every twO nonadacent vertices x and y, G + cy is induced
matching extendable. In chaPter 3, we charaCteri2e the mhanal IM-unextend-
Theorem 2. 1 (Tutte's theorem) A graPh G has a perfect matching if
and ouly if for every S= V(G), o(G -- S) I S I. Here, o(H) is the num-
ber of edd componentS of H.