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* Princeton Discrete Math Seminar *
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Date: Thursday 21st September, 3:00 in Fine Hall 224.
Speaker: Jie Ma (University of Science and Technology of China)
Title: Stability results in graphs of given circumference
In this talk we will discuss some Turan-type results on graphs with
a given circumference. Let W_{n,k,c} be the graph obtained from a
clique K_{c-k+1} by adding n-(c-k+1) isolated vertices each joined
to the same k vertices of the clique, and let f(n,k,c)=e(W_{n,k,c}).
Improving the Erdos-Gallai theorem, Kopylov proved in 1977 that for
c max (f(n,3,c),f(n,[c/2]-1,c) ), then
either G is a subgraph of W_{n,2,c} or W_{n,[c/2],c}, or c is odd
and G is a subgraph of a member of two well-characterized families
which we define as X_{n,c} and Y_{n,c}. We extend and refine their
result by showing that if G is a 2-connected graph on n vertices with
minimum degree at least k and circumference c such that 9 < c < n and
e(G) > max (f(n,k+1,c),f(n,[c/2]-1,c)), then one of the following holds:
(i) G is a subgraph of W_{n,k,c} or W_{n,[c/2],c},
(ii) k=2, c is odd, and G is a subgraph of a member of
X_{n,c} cup Y_{n,c}, or
(iii) k > 2 and G is a subgraph of the union of a clique
K_{c-k+1} and some cliques K_{k+1}'s, where any two cliques share the
same two vertices.
This provides a unified generalization of the above result of
Furedi et al. as well as a recent result of Li et al. and independently,
of Furedi et al. on non-Hamiltonian graphs. Moreover, we prove a
stability result on a classical theorem of Bondy on the circumference.
We use a novel approach, which combines several proof ideas including
a closure operation and an edge-switching technique.
Next week: Ron Aharoni
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