In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of Kn, there is a rainbow path on (3/4− o(1))n vertices, improving on the previously best bound of (2n + 1)/3 from [?]. Similarly, a k-rainbow path in a proper...

One consequence of an old conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose G is a multigraph with maximum degree ∆, such that no vertex subset S of odd size at most ∆ induces more than (∆+1)(|S|−1)/2 edges. Then G has an edge coloring with ∆ + 1 colors. Here we prove a weakened version of this statement.

The edge strength s(G) of a multigraph G is the minimum number of colors in a minimum sum edge coloring of G. We give closed formulas for the edge strength of bipartite multigraphs and multicycles. These are shown to be classes of multigraphs for which the edge strength is always equal to the chromatic index.

An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph induced by any two color classes is acyclic. The acyclic chromatic index, χa(G), is the smallest number of colors allowing an acyclic edge-coloring of G. Clearly χa(G) ≥ ∆(G) for every graph G. Cohen, Havet, and Müller conjectured that there exists a constant M such that every planar graph with ∆(G) ≥M h...

The problem of edge coloring a bipartite graph is to color the edges so that adjacent edges receive di erent colors An optimal algorithm uses the minimum number of colors to color the edges We consider several approximation algorithms for edge coloring bipartite graphs and show tight bounds on the number of colors they use in the worst case We also present results on the constrained edge colori...

In this paper, we study a new coloring parameter of graphs called the gap vertexdistinguishing edge coloring. It consists in an edge-coloring of a graph G which induces a vertex distinguishing labeling of G such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distingu...

Given a graph G and an edge coloring C of G, a heterochromatic cycle of G is a cycle in which any pair of edges have distinct colors. Let dc(v), named the color degree of a vertex v, be the maximum number of distinct colored edges incident with v. In this paper, some color degree conditions for the existence of heterochromatic cycles are obtained.

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 ...