# Stability results in graphs of given circumference

Thursday, September 21, 2017 -

3:00pm to 4:30pm

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<n, any 2-connected graph G on n vertices with circumference c has at most max (f(n,2,c),f(n,[c/2],c)) edges, with equality if and only if G equals W_{n,2,c} or W_{n,[c/2],c}. Recently, Furedi et al. obtained a stability version of Kopylov's theorem. They proved that if G is a 2-connected graph on n vertices with circumference c such that 9 < c < n and e(G) > 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.

Speaker:

Jie Ma

University of Science and Technology of China

Event Location:

Fine Hall 224