Abstract In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. same article, authors proved some results on solvability Cartesian products, given solvable or distance 2-solvable We extend these products certain unsolvable particular, we prove that ladders grid graphs are and, further, even product two stars, which in a sense “most”

Abstract The general position number $$\mathrm{gp}(G)$$ gp ( G ) of a connected graph G is the cardinality largest set S vertices such that no three distinct from lie on common geodesic; sets are refereed to as gp-sets . cylinders $$P_r...

There are four standard products of graphs: the direct product, the Cartesian product, the strong product and the lexicographic product. The chromatic number turned out to be an interesting parameter on all these products, except on the Cartesian one. A survey is given on the results concerning the chromatic number of the three relevant products. Some applications of product colorings are also ...

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspect...