Vertex colorings with a distance restriction

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

Strong parity vertex coloring of plane graphs

Czap and Jendrol’ introduced the notions of strong parity vertex coloring and the corresponding strong parity chromatic number χs. They conjectured that there is a constant bound K on χs for the class of 2-connected plane graphs. We prove that the conjecture is true with K = 97, even with an added restriction to proper colorings. Next, we provide simple examples showing that the sharp bound is ...

On perfect colorings of the distance-2 graph of the 24-cube

A vertex 2-coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly i vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist if c = 3, 6, 8, 9,...

D-strong Total Colorings of Graphs

If c : V ∪ E → {1, 2, . . . , k} is a proper total coloring of a graph G = (V,E) then the palette S[v] of a vertex v ∈ V is the set of colors of the incident edges and the color of v: S[v] = {c(e) : e = vw ∈ E} ∪ {c(v)}. A total coloring c distinguishes vertices u and v if S[u] 6= S[v]. A d-strong total coloring of G is a proper total coloring that distinguishes all pairs of vertices u and v wi...

DISTANCE k COLORINGS OF HAMMING GRAPHS

A coloring of the vertices of a graph G is a distance k coloring of G if and only if any two vertices lying on a path of length less than or equal to k are given different colors. Hamming graphs are Cartesian (or box) products of complete graphs. In this paper, we will consider the interaction between coding theory and distance k colorings of Hamming graphs.