The maximum number of cliques in graphs with bounded odd circumference

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Tur\'an-type result is an extension of the celebrated Erd\H{o}s and Gallai theorem and a strengthening of Luo's recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.


Introduction
A central topic of extremal combinatorics is to investigate sufficient conditions for the appearance of given substructures.In particular, for a given graph H and a set of graph F, the generalized Turán number ex(n, H, F) denotes the maximum number of copies of H in a graph on n vertices containing no F as a subgraph, for every F ∈ F. In 1959, Erdős and Gallai [1] determined the maximum number of edges in a graph with a small circumference.For every integers n and k such that n ≥ k ≥ 3, they proved where K k denotes the complete graph with k vertices and C ≥k denotes the family of cycles of length at least k.The bound is sharp for every n congruent to one modulo k − 2. The equality is attained by graphs with n−1 k−2 maximal 2-connected blocks each isomorphic to K k−1 .Recently Li and Ning [2] proved that in order to obtain the same upper-bound it is enough to forbid only long even cycles where C even ≥2k denotes the family of even cycles of length at least 2k, that is {C 2k , C 2k+2 , . . .}.We also denote the family of odd cycles of length at least 2k + 1 by C odd ≥2k+1 := {C 2k+1 , C 2k+3 , . . .}.Note that by considering the complete balanced bipartite graph, one can not achieve a linear bound for edges in graphs with a small odd circumference.On the other hand, Voss and Zuluga [5] proved that every 2-connected graph G with minimum degree at least k ≥ 3, with at least 2k + 1 vertices, contains an even cycle of length at least 2k.Even more, if G is not bipartite then it contains an odd cycle of length at least 2k − 1.
The celebrated Erdős and Gallai theorem is well studied and well understood.There are numerous papers strengthening, generalizing, and extending for different classes of graphs.Recently Luo [3] for all n ≥ k ≥ 4.This bound is a great tool for obtaining results in hypergraph theory.In order to highlight its significance, that bound was consequently reproved with different methods multiple times [4,6].In this paper, we strengthen Luo's theorem.In particular, we obtain the tight same bounds for graphs with bounded odd circumferences.On the other hand, the result for graphs with bounded even circumference is trivial after applying Equation 1 and the following easy corollary of Equation 2, where P k+1 denotes the path of length k.Since the graph G does not contain a cycle of length 2k, for each vertex the neighborhood contains no path of length 2k − 2. In particular, for all r ≥ 3 we have For graphs with small odd circumferences, we have the following theorem.

The equality holds if and only if
It is interesting to ask for which integers and for which congruence classes the same phenomenon still holds.In this direction, we would like to rise a modest conjecture.
Conjecture 1.For an integer k ≥ 2 and a sufficiently large n.Let G be an n vertex C 3ℓ+1 -free graph for every integer ℓ ≥ k.Then for every r ≥ 2, the number of cliques of size r in G is at most The equality holds if and only if n − 1 is divisible by 3k − 1 for a connected n-vertex graph which consists of n−1 3k−1 maximal 2-connected blocks isomorphic to K 3k .
Moreover, we would like to propose the following conjecture.Let p be a prime number and C prime ≥p be the family of all cycles of length at least p.
Conjecture 2. For integers n, r and a prime p satisfying r < p, we have The equality holds if and only if n − 1 is divisible by p − 2, for a connected n-vertex graph which consists of n−1 p−2 maximal 2-connected blocks isomorphic to K p−1 .

Proof of the main result
Proof.We prove Theorem 1 by induction on the number of vertices of the graph.The base cases for n ≤ 2k are trivial.Let G be a graph on n vertices where n > 2k.We assume that every C odd ≥2k+1 -free graph on m vertices, for some m < n, contains at most m−1 2k−1 2k r copies of K r and the equality is achieved for the class of graphs described in the statement of the theorem.
If δ(G), the minimum degree of G, is at most k + 2 then we are done by the induction hypothesis since k ≥ 6.From here, we may assume G is a graph with δ(G) > k + 2, and each edge of G is in a copy of K r .Let v 1 v 2 v 3 . . .v m be a longest path of G such that v 1 is adjacent to v t and t is the maximum possible among the longest paths.
Proof.Consider a family P contains all longest paths of G on the vertex set {v 1 , v 2 , v 3 , . . ., v m } such that v t v t+1 . . .v m is a sub-path of them.Let T 1 be the set of terminal vertices, excluding v m , of paths from P.
. ., u t }.By the minimality of r, the vertex u 1 is not adjacent to two consecutive vertices from {u r+1 , u r+2 , . . ., v t }.Since if v 1 is adjacent to u i and u i+1 for r + 1 ≤ i ≤ t, then forms a path from P, contradicting the minimality of r.
Proof.Suppose otherwise, t > 2k.Let C denote the cycle v 1 v 2 . . .v t v 1 .Since G is C odd ≥2k+1 -free, t is even and vertices v t−3 and v t−2 have no common neighbors in G[V (G) \ V (C)].On the other hand, since every edge is in a K r there is a vertex v l incident with both v t−3 and v t−2 .Let us denote c := t − 2k.Since G is C odd ≥2k+1 -free, c − 1 ≤ l ≤ 2k − 4 holds.We take indices modulo t in this claim.
There is no i such that v i is adjacent with v t−1 and v i+1 is adjacent with v 1 .Since otherwise, the following cycle is odd with a length greater than 2k, There is no such i such that both l < i < l + c − 2 and v 1 v i is an edge of G hold.Since otherwise, one of the following cycles is an odd cycle longer than 2k, Similarly, there is no i such that l < i < l + c and v t−1 v i is an edge of G. Finally, we have N (v 1 ) ∩ {v l+2 , . . ., v l+c−3 } = ∅.