An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f . In th...

The graph coloring problem is a notoriously hard problem, for which we do not have efficient algorithms. A coloring of a graph is an assignment of colors to its vertices such that the end points of every edge have different colors. A k-coloring is a coloring that uses at most k distinct colors. The graph coloring problem is to find a coloring that uses the minimum number of colors. Given a 3-co...

Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, i.e., partition the vertices to batches which are independent sets, and to assign to each batch a contiguous set of colors, whose size equals to the maximum color requirement of any vertex ...

Let us improve this bound. Assume that G is a connected graph and T is its spanning tree rooted at r. Let us consider an ordering of V (G) in which each vertex v appears after its children in T . Now, for v 6= r we have |N(vi) ∩ {v1, . . . , vi−1}| ≤ deg v − 1, so c(vi) ≤ deg vi for vi 6= r. Unfortunately, the greedy may still need to use ∆(G) + 1 colors if deg r = ∆(G) and each child of r happ...

In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighbor...