Thesis title: About Second Neighborhood and Out-Domination in Tournaments, and Edge Packing Coloring of Graphs.

My thesis aims to investigate two separate research topics by working on open problems within each:

Topic 1: Second Neighborhood and Out-Domination in Tournaments.

The second out-degree of a vertex x, denoted by d⁺⁺(x), is the number of vertices y such that d⁺(x, y) = 2, where d⁺(x, y) is the length of the shortest xy-directed path, if it exists. It is obvious that the sum of the first out-degrees of the vertices in a digraph is nothing but the number of its arcs. Unlike the first out-degree, the summation of the second out-degrees of the vertices in a digraph is not constant with respect to the number of vertices and arcs. In this thesis, we characterize, as a function of some integer n, the values that can be the summation of the second out-degrees of the vertices in a tournament of order n. We use the new concept of king degree in order to settle the problem. A king is a vertex x such that d⁺(x,y) ≤ 2 for all vertices y. The king degree of a vertex x, denoted by k⁺(x), is the number of vertices that can be reached from x by a directed path of length at most 2. Denote by k⁺(T) the summation of the king degrees of all vertices of T. We are working on several open problems introduced in this area, including the following: What is the number of tournaments T of order n such that k⁺(T)=k? Can we characterize these tournaments?

Topic 2: Edge Packing Coloring of Graphs.

Given a non-decreasing sequence S = (s₁,s₂,...,s_k) of positive integers, an S-packing edge-coloring of a graph G is an assignment of colors (s₁,s₂,...,s_k) to the edges of G such that the edges at distance at least s_i+1 are colored by s_i. It is an extension of the proper edge-coloring. In this thesis, we are going to treat the problem of packing edge-coloring first for subcubic graphs, where there are still a lot of open problems, and then for graphs whose maximum degree is greater than three, trying to generalize some results concerning the edge packing coloring of subcubic graphs. Here are some conjectures we are working on: Is every subcubic graph (1,1,2,2,2,2)-packing edge-colorable? Is every 2-saturated subcubic graph (1,1,2,2)-packing edge-colorable?

Aya Alhussein, Ayman El Zein, The King Degree and the Second Out-Degree of Tournaments, Discrete Math. 348 (2025) 114497.

Delivered a Seminar on Graph Theory titled by "The King Degree and the Second Out-Degree of Tournaments", on 18 April 2024, at the Lebanese University-Faculty of Sciences, Beirut-Hadath, Department of Mathematics. Abstract: In this talk, we characterize, as a function of some integer n, the values that can be the summation of the second out-degrees of the vertices in a tournament of order n. We use the new concept of king degree in order to settle the problem.