b-Chromatic Number (χ₍b₎ (G)): The maximum number of colours for which a graph G can be b-coloured. Proper Colouring: A colouring of the vertices of a graph in which no two adjacent vertices ...

Abstract The adjacent vertex-distinguishing total chromatic number of a graph 𝐺, denoted by χ𝑎𝑡 (𝐺), is the smallest denoted by 𝑘 for which 𝐺 has a proper total 𝑘-coloring such that any two ...

The Hadwiger-Nelson problem is a bit different. Instead of considering a finite number of vertices, as there would be on a map, it considers infinitely many vertices, one for each point in the plane.

Abstract.For two vertices 𝑢 and 𝑣 of a graph 𝐺, the set 𝐼 [𝑢, 𝑣] consists of all vertices lying on some 𝑢 - 𝑣 geodesic in 𝐺. If 𝑆 is a set of vertices of 𝐺, then 𝐼 [𝑆] is the union of all ...

The b-chromatic number is a refined measure in which, beyond achieving a proper colouring, each colour class must contain at least one vertex, known as a b-vertex, with adjacent vertices spanning ...