Cliques and Constructors in “Hats” Game. I

The Hats game. On maximum degree and diameter

“…We call constructors the theorems, which allow us to build new winning graphs by combining the graphs, for which their winning property is already proved. Here are several constructors from the papers [9,11,2]. E 2 ) be two graphs, S ⊆ G 1 be a clique, and v ∈ V 2 .…”

“…This theorem was proven in [2, Lemma 7]. For partial case where S is a single vertex, a sum of Hats games was considered in [9] where it is called a product of games. We will use this operation often so let us described it in more details.…”

“…A product of games G 1 , G 2 with respect to vertex A is just a ({A}, v)-sum of G 1 and G 2 . We will denote product of games by more elegant notation [9]. For vectors and functions on V we will use the sign…”

“…In their previous works [9] and [11] (joint with V. Retinsky), the authors consider the version of the game with variable number of hats (i.e., the number of possible colors can be different from sage to sage). This version of game is not only of its own interest, but opens a more flexible approach to the classic Hats game analysis.…”

“…The rest of the paper is organized as follows. In section 2 we give necessary definitions and notations, recall several theorem-constructors from [9,11,2]. In section 3 in terms of independence polynomials we define maximal games and prove theorem that connects maximal games with constructors.…”