Open standard CMO for parametric modelling based on semantic web

Connectedness and Hamiltonicity of graphs on vertex colorings

“…Thus we may assume without loss of generality that φ(x) = φ ′ (x) = 1. Let π ∈ V (G); we find ρ ∈ V (G) − {π} such that there is a Hamiltonian path through G from π to ρ, with ρ(u) = φ ′ (x) and ρ(v) = φ ′ (y) (allowing φ ′ to be extended to some proper k-coloring ρ ′ of H by coloring uv like ρ does, so ρ and ρ ′ will be adjacent in G 1 4 (H) since they only differ on w).…”

“…We complete our Hamiltonian cycle through G 1 4 (H) by first taking our path through G b−2 and G b−1 , then setting α b as the coloring in V (G b ) that agrees β b−1 on u and v (such an α b exists because φ b (x) = 1 and φ b (y) = 2 while β b−1 colors u from {2, 3, 4} and colors v from {3, 4}, and β b−1 and α b are adjacent in G 1 4 (H) because they only differ on the vertex of H ′ where φ b−1 and φ b differ), and finally finding a Hamiltonian path through G b (such a path exists: if β b uses a color outside of {3, 4} on u or v, then we selected α b−1 to color u and v from {3, 4}, so there exists a Hamiltonian path through G b from α b−1 to any other vertex; if β b colors u and v from {3, 4}, then we selected α b−1 to color u with 2 and v from {3, 4}, so there exists a Hamiltonian path through G b from α b−1 to any coloring that colors u and v from {3, 4}).…”

“…The mixing number of H, denoted k 1 (H), is the least K such that G k (H) is connected for all k ≥ K. In 2008, Cereceda, van den Heuvel, and Johnson [2] studied k 1 (H). In particular, they showed that k 1 (H) ≤ d + 2 if H is d-degenerate, meaning every subgraph of H has a vertex of degree at most d. Other papers on the connectedness of G k (H), or on finding paths between particular vertices of G k (H), include [1], [3], and [4].…”