Total and Strong Edge Colorings on Human Chain Network

Edge colorings and total colorings of integer distance graphs

r-Strong edge colorings of graphs

If c : E → {1, 2, . . . , k} is a proper edge 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: S(v) = {c(e) : e = vw ∈ E}. An edge coloring c distinguishes vertices u and v if S(u) 6= S(v). A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance d(u, v) ≤ d. The minimu...

Strong edge colorings of uniform graphs

For a graph G = (V (G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in [3], [4], [12], and [16]. In this paper, we consider χs(G) for a related class of graphs ...

Strong edge-colorings for k-degenerate graphs

We prove that the strong chromatic index for each k-degenerate graph with maximum degree ∆ is at most (4k − 2)∆ − k(2k − 1) + 1. A strong edge-coloring of a graph G is an edge-coloring so that no edge can be adjacent to two edges with the same color. So in a strong edge-coloring, every color class gives an induced matching. The strong chromatic index χs(G) is the minimum number of colors needed...

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...