We discuss the P versus NP problem from the perspective of addition operation about polynomial functions. Two contradictory propositions for the addition operation are presented. With the proposition that the sum of k (k ≤ n) polynomial functions on n always yields a polynomial function, we prove that P = NP , considering the maximum clique problem. And with the proposition that the sum of k po...

We prove that, unless any problem in NP can be solved in probabilistic polynomial time, for any ǫ > 0, the size of the largest clique in a graph with n nodes is hard to approximate in polynomial time within a factor n. This is done by constructing, for any δ > 0, a probabilistically checkable proof for NP which uses logarithmic randomness and δ amortized free bits.

We provide an extensive study of the differential properties of the functions x 7→ x t −1 over F2n , for 1 < t < n. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x t + bx + (b + 1)x where b varies in F2n .We prove a strong relationship between the differential spectra of x 7→ x t −1 and x 7→ x s −1 for s = n− t+ ...

A permutation π contains a permutation σ as a pattern if it contains a subsequence of length |σ| whose elements are in the same relative order as in the permutation σ. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption NP 6⊆ co-NP/poly) by introducing a new polynomial reduction...

Recently, Hazan and Krauthgamer showed [12] that if, for a fixed small ε, an ε-best ε-approximate Nash equilibrium can be found in polynomial time in two-player games, then it is also possible to find a planted clique in Gn,1/2 of size C log n, where C is a large fixed constant independent of ε. In this paper, we extend their result to show that if an ε-best ε-approximate equilibrium can be eff...

In this note it is proved that if a graph G of order n has an irreducible covering of its vertex set by n− k cliques, then its clique number ω(G) ≤ k + 1 if k = 2 or 3 and ω(G) ≤ ( k bk/2c) if k ≥ 4. These bounds are sharp if n ≥ k + 1 (for k = 2 or 3) and n ≥ k + ( k bk/2c) (for k ≥ 4).

In 2022, the idea of a bivariate polynomial which represents number complete subgraphs graph with corresponding common neighborhood systems has been introduced in [3]. present work, we extend this notion to more restricted case by considering maximal connected cliques given graph. Besides characterizing corona two graphs, establish clique resulting from graphs. Received: June 2, 2023Accepted: J...

Directed edge path graphs are the intersection graphs of directed paths in a directed tree, viewed as sets of edges. They were studied by Monma and Wei (J. Comb. Theory B 41 (1986) 141–181) who also gave a polynomial time recognition algorithm. In this work, we show that the clique graphs of these graphs are exactly the two sections of the same kind of path families, and give a polynomial time ...