CHROMATICALLY EQUIVALENT k-BRIDGE HYPERGRAPHS

An h-uniform hypergraph (h ≥ 2) H = (V, E) of order n = |V| and size m = |E|, consists of a vertex set V(H) = V and edge set E(H) = E , where E ⊂ V and |E| = h for each edge E in E . H is said to be linear if 0 ≤ |E ∩ F | ≤ 1 for any two distinct edges E,F ∈ E(H) [1]. Let P h,1 p denote the linear path consisting of p ≥ 1 edges E1, . . . , Ep such that |E1| = . . . = |Ep| = h, |Ek ∩ El| = 1 if {k, l} = {i, i + 1} for any 1 ≤ i ≤ p− 1 and 0 otherwise. Each vertex from E1 \E2 or from Ep \Ep−1 will be called an end vertex of P h,1 p . For any positive integers a1, . . . , ak ∈ N and h≥2 we denote by θ(h; a1, . . . , ak) the h-uniform linear hypergraph consisting of k linear paths P h,1 a1 , P h,1 a2 , . . . , P h,1 ak of lengths a1, a2, . . . , ak respectively, having in common only two fixed ends. It is a parallel hypergraph [4] of order (h − 1)(a1 + . . . + ak) − k + 2. For example, θ(3; 4, 3, 5) is depicted in Fig. 1 with x and y common ends of the paths. This linear hypergraph will be called a k-bridge hypergraph; θ(2; a1, . . . , ak) is a notation for a k-bridge graph (see [3]). A λ-coloring of a hypergraph H is a function f : V(H)→ {1, . . . , λ} such that each edge E ∈ E(H) contains two vertices x and y having different colors f(x) 6= f(y). The number of λ-colorings of H is given by a polynomial P (H,λ)