Bounds on Edge Colorings with Restrictions on the Union of Color Classes

We consider constrained proper edge colorings of the following type: Given a positive integer j and a family F of connected graphs on 3 or more vertices, we require that the subgraph formed by the union of any j color classes has no copy of any member of F . This generalizes some well-known types of colorings such as acyclic edge colorings, distance-2 edge colorings, low treewidth edge colorings, etc. For such a generalization of restricted colorings, we obtain an upper bound of O(d) on the minimum number of colors sufficient for such a coloring. Here, d refers to the maximum degree of the graph and θ is a parameter defined by θ = θ(j,F) = maxH∈F (|V (H)|−2) (|E(H)|−j) . Our proof is based on probabilistic arguments. In particular, we obtain O(d) upper bounds for proper edge colorings with various interesting restrictions placed on the union of color classes. For example, we obtain O(d) upper bounds on edge colorings with restrictions such as (i) the union of any 3 color classes should be an outerplanar graph, (ii) the union of any 4 color classes should have treewidth at most 2, (iii) the union of any 5 color classes should be planar, (iv) the union of any 16 color classes should be 5-degenerate, etc. We also consider generalizations where we require simultaneously for several pairs (ji,Fi) (i = 1, . . . , s) that the union of any ji color classes has no copy of any member of Fi and obtain upper bounds on the corresponding chromatic indices. As a corollary, we obtain that each of the four restrictions above can be satisfied simultaneously using O(d) colors. Proposed running head : Bounds on restricted edge colorings.