### Proper mean colorings of graphs

Gary Chartrand, James Hallas, Ping Zhang
2020 Discrete Mathematics Letters
For an edge coloring of a connected graph G of order 3 or more with positive integers, the chromatic mean of a vertex v of G is the sum of the colors of the edges incident with v divided by the degree of v. Only edge colorings c are considered for which the chromatic mean of every vertex is a positive integer. If adjacent vertices have distinct chromatic means, then c is a proper mean coloring of G. The maximum vertex color in a proper mean coloring c of G is the proper mean index of c and the
more » ... index of c and the proper mean index µ(G) of G is the minimum proper mean index among all proper mean colorings of G. The proper mean index is determined for complete graphs, cycles, stars, double stars, and paths. The non-leaf minimum degree δ * (T ) of a tree T is the minimum degree among the non-leaves of T . It is shown that if T is tree with δ * (T ) ≥ 10 or a caterpillar with δ * (T ) ≥ 6, then µ(T ) ≤ 4. Furthermore, it is conjectured that χ(G) ≤ µ(G) ≤ χ(G) + 2 for every connected graph G of order 3 or more.