Some Properties Of A Class Of Claw-free Graphs

In this thesis ,we mainly study the properties of a class of claw-free graphs, and we have the following results:(1) IfG is a connected claw-free graph , M ( G )={ x | x∈V ( G), x is locally connected } is a dominating set of G , M ( G )has two components M 1 ,M 2, then cl ( G )is completed if and only if there is a cycleC inG ,andC joins these two components such that there is a locally disconnected vertex x satisfies d_G1 (x)≥2, d_G2(x)≥2, G_i = i)∪N (M_i)>,d_Gi (x) = |N_G (x)∩V(G_i)|.(2) IfG is a connected claw-free graph , M ( G )={ x | x∈V ( G), x is locally connected } is a dominating set of G , M ( G )has three components M₁ , M₂ ,M₃, then cl ( G ) is completed if and only ifG satisfies one of the following conditions:(i) there is a cycleC inG ,andC joins these three components such that there are three locally disconnected vertices ; (ii) there are at least two componentsM_i1 ,M_i2 such that V(G_i) 1∪V Gi2satisfies the conditions of (i) and there is a cycle C in G ,and C joins these three components such that there is a locally disconnected vertex x satisfies ,d_<VG_i1∪VG_i2x>≥2d_Gi3x≥2, Gi = V(M_i)∪N (M_i) , d G_i(x) = NG (x)∩VG_i,at last,we extend this result to M ( G ) has r components.(3) IfG is a connected claw-free graph , M ( G )={ x | x∈V ( G). x is locally connected }is a dominating set of G , M ( G )has two components ,if cl (G )is completed ,thenG is pancyclic.(4) IfG is a connected claw-free graph , M ( G )={ x | x∈V ( G), x is locally connected }is a dominating set ofG , M ( G ) has three components ,if cl ( G )is completed ,thenG is pancyclic except one case.(5) we get a new lower bound of cm ( n ), cm ( n )is the cycle lengths that can be missing in a claw-free on n vertices with complete closure.(6) IfG is a claw-free graph, G≥1 0,G have at least a vertex x , ( )N Gx is not locally connected or N G(x) is completed , M ( G )={ x | x∈V ( G), x is locally connected }is a dominating set of G , and M ( G )is connected, thenG has 2-factor with two components, moreover, G≥1 0is the best possible .