Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G. The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisn...

Many symmetric properties are well-explored in graph theory, especially coloring, such as graphs defined by the automorphism groups, drawing of planar graphs, and functions which used to count number specific colorings a graph. This paper is devoted studying star edge coloring 1-planar graphs. The chromatic index χst′(G) G smallest k for edges can be colored using colors so that no two adjacent...