5 research outputs found
Generalized Tur\'an problems for even cycles
Given a graph and a set of graphs , let
denote the maximum possible number of copies of in an -free
graph on vertices. We investigate the function , when
and members of are cycles. Let denote the cycle of
length and let . Some of our main
results are the following.
(i) We show that for any .
Moreover, we determine it asymptotically in the following cases: We show that
and that the maximum
possible number of 's in a -free bipartite graph is .
(ii) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds,
then for any we have . We prove that forbidding any other even cycle
decreases the number of 's significantly: For any , we have
More generally,
we show that for any and such that , we have
(iii) We prove provided a
strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true
when ). Moreover, forbidding one more cycle decreases the number
of 's significantly: More precisely, we have and for .
(iv) We also study the maximum number of paths of given length in a
-free graph, and prove asymptotically sharp bounds in some cases.Comment: 37 Pages; Substantially revised, contains several new results.
Mistakes corrected based on the suggestions of a refere
Subgraph densities in a surface
Given a fixed graph that embeds in a surface , what is the
maximum number of copies of in an -vertex graph that embeds in
? We show that the answer is , where is a
graph invariant called the `flap-number' of , which is independent of
. This simultaneously answers two open problems posed by Eppstein
(1993). When is a complete graph we give more precise answers.Comment: v4: referee's comments implemented. v3: proof of the main theorem
fully rewritten, fixes a serious error in the previous version found by Kevin
Hendre
Counting copies of a fixed subgraph in F-free graphs
Fix graphs F and H and let ex(n, H, F) denote the maximum possible number
of copies of the graph H in an n-vertex F-free graph. The systematic study of this
function was initiated by Alon and Shikhelman [J. Comb. Theory, B. 121 (2016)]. In
this paper, we give new general bounds concerning this generalized Tur´an function.
We also determine ex(n, Pk, K2,t) (where Pk is a path on k vertices) and ex(n, Ck, K2,t)
asymptotically for every k and t. We also characterize the graphs
F that cause the function ex(n, Ck, F) to be linear in n. In the final section we discuss
a connection between the function ex(n, H, F) and Berge hypergraph problem