Alon, Krech, and Szabó [SIAM J. Discrete Math., 21 (2007), pp. 66–72] called an edge-coloring of a hypercube with p colors such that every subcube of dimension d contains every color a d-polychromatic p-coloring. Denote by pd the maximum number of colors with which it is possible to d-polychromatically color any hypercube. We find the exact value of pd for all values of d.

Let G be a simple, undirected graph. We say that two edges of G are within distance 2 of each other if either they are adjacent or there is some other edge that is adjacent to both of them. A distance-2-edge-coloring of G is an assignment of colors to edges so that any two edges within distance 2 of each other have distinct colors, or equivalently, a vertex-coloring of the square of the line gr...

The adaptable chromatic number of a multigraph G, denoted χa(G), is the smallest integer k such that every edge labeling of G from [k] = {1, 2, · · · , k} permits a vertex coloring of G from [k] such that no edge e = uv has c(e) = c(u) = c(v). Hell and Zhu proved that for any multigraph G with maximum degree ∆, the adaptable chromatic number is at most lp e(2∆− 1) m . We strengthen this to the ...

Given a graph G and an edge coloring C of G, an alternating cycle of G is such a cycle of G in which any adjacent edges have distinct colors. Let dc(v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, some color degree conditions for the existence of alternating cycles of length 3 or 4 are obtained. We al...

A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . We show that the sum choice number of a 2 × n array (equivalent to list edge coloring K2,n and to list vertex coloring the cartesian product K22Kn) is n2 ...

An edge coloring of a graph G is called Mi-edge coloring if at most i colors appear at any vertex of G. Let Ki(G) denote the maximum number of colors used in an Mi-edge coloring of G. In this paper we determine the exact value of K2(G) for any graph G of maximum degree at most 3. Mathematics Subject Classification: 05C15, 05C38

An edge coloring φ of a graph G is called an Mi-edge coloring if |φ(v)| ≤ i for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let Ki(G) denote the maximum number of colors used in an Mi-edge coloring of G. In this paper we establish some bounds of K2(G), present some graphs achieving the bounds and determine exact values of K2(G) for dense graphs.

An r-edge-coloring of a graph is an assignment of r colors to the edges of the graph. An exactly r-edge-coloring of a graph is an r-edge-coloring of the graph that uses all r colors. A matching of an edge-colored graph is called rainbow matching, if no two edges have the same color in the matching. In this paper, we prove that an exactly r-edge-colored complete graph of order n has a rainbow ma...