Let k be a positive integer. A k ‐circular‐ L (2, 1)‐labelling of a graph G is an assignment f from V ( G ) to {0, 1, …, k −1} such that, for any two vertices u and v , | f ( u ) − f ( v )| k ≥ 2 if u and v are adjacent, and | f ( u ) − f ( v )| k ≥ 1 if u and v are at distance 2, where | x | k = min{| x |, k −| x |}. The minimum k such that G admits a k ‐circular‐ L (2, 1)‐labelling is called the circular‐ L (2, 1)‐labelling number (or just the σ ‐number) of G , denoted by σ ( G ). The exact values of σ ( P m ⊠ C n ) and σ ( C m ⊠ C n ) for some m and n have been determined in this study. Finally, it has been concluded that σ ( C m ⊠ C n ) ≤ 13 for n ≥ m ≥ 220.

Let u and v be vertices of a graph G = ( V , E ) and d ( u , v ) be the distance between u and v in G . For positive integers k 1 , k 2 , … , k n with k 1 > k 2 >⋯> k n an L ( k 1 , k 2 , … , k n )‐labelling of G is a function f : V ( G ) → {0, 1, … } such that for every u , v ∈ V ( G ) and for all 1 ≤ i ≤ n , |f ( u ) − f ( v ) | ≥ k i if d ( u , v ) = i . The span of f is the difference between the largest and the smallest numbers in f ( V ( G )). The , k 2 ,…, k n ‐number of G is the minimum span over all L ( k 1 , k 2 , … , k n )‐labellings of G . In this study, an integer linear programming model and a satisfiability test reduction for an L ( k 1 , k 2 , … , k n )‐labelling are proposed. Both approaches are used for studying the λ 3,2,1 ‐numbers of strong, Cartesian and direct products of paths and cycles.

We present several new constructive lower bounds for classical Ramsey numbers. In particular, the inequality $R(k,s+1) \geq R(k,s)+2k-2$ is proved for $k \geq 5$. The general construction permits us to prove that, for all integers k, l, with $k \geq 5$ and $l \geq 3$, the connectivity of any Ramsey-critical $(k,l)$-graph is at least k, and if $k \geq l-1 \geq 1$, $k \geq 3$ and $(k,l) \neq (3,2)$, then such graphs are Hamiltonian. New concrete lower bounds for Ramsey numbers are obtained, some with the help of computer algorithms, including: $R(5,17) \geq 388$, $R(5,19) \geq 411$, $R(5,20) \geq 424$, $R(6,8) \geq 132$, $R(6,12) \geq 263$, $R(7,8) \geq 217$, $R(7,9) \geq 241$, $R(7,12) \geq 417$, $R(8,17) \geq 961$, $R(9,10) \geq 581$, $R(12,12) \geq 1639$, and also one three-color case $R(8,8,8) \geq 6079$.

In this paper, a new genetic algorithm for solving multi-constrained optimization problems based on KS function is proposed. Firstly, utilizing the agglomeration features of KS function, all constraints of optimization problems are agglomerated to only one constraint. Then, we use genetic algorithm to solve the optimization problem after the compression of constraints. Finally, the simulation results on benchmark functions show the efficiency of our algorithm.