In this paper we show a natural extension of the idea used by Illés, Szőnyi and Wettl which proved that the flags of PG(2, q) can be partitioned into (q−1) √ q+3q strong representative systems for q an odd square. From a generalization of the Buekenhout construction of unitals their idea can be applied for any non-prime q to yield that q +2q strong representative systems partition the flags of ...

In this short article, we shall try to find an explicit formula for Maslov index of triangles joining intersections points of three half-dimensional tori in the symmetric product of a surface. The method will also yield a formula for the intersection number of such a triangle with the flat diagonal in the symmetric product.

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (∆ + 1)vertex coloring and (2∆ − 1)-edge coloring in a graph with maximum degree ∆. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following th...

A strong edge-coloring of a graph is a function that assigns to each edge a color such that every two distinct edges that are adjacent or adjacent to a same edge receive different colors. The strong chromatic index χs(G) of a graph G is the minimum number of colors used in a strong edge-coloring of G. From a primal-dual point of view, there are three natural lower bounds of χs(G), that is σ(G) ...

Graph coloring is a central problem in distributed computing. Both vertexand edge-coloring problems have been extensively studied in this context. In this paper we show that a (2∆ − 1)-edge-coloring can be computed in time smaller than log n for any > 0, specifically, in e √ log logn) rounds. This establishes a separation between the (2∆ − 1)-edge-coloring and Maximal Matching problems, as the ...

An injective $k$-edge-coloring of a graph $G$ is an assignment colors, i.e. integers in $\{1, \ldots , k\}$, to the edges such that any two each incident with one distinct endpoint third edge, receive colors. The problem determining whether $k$-coloring exists called k-INJECTIVE EDGE-COLORING. We show 3-INJECTIVE EDGE-COLORING NP-complete, even for triangle-free cubic graphs, planar subcubic gr...

This file contains the proofs of Theorems 1.2 and 1.3.