iii Acknowledgments I want to thank my thesis advisor Jörg Rothe for all of his support during the past four years. In the first place, I am deeply grateful to him for giving me the chance to be part of his research team. Without his great efforts, I would never have had the chance to work in the scientific community. Many inspiring and valuable discussions with him initiated fruitful ideas tha...

Assume that each vertex of a graph G is assigned a constant number q of nonnegative integer weights, and that q pairs of nonnegative integers li and ui, 1 ≤ i ≤ q, are given. One wishes to partition G into connected components by deleting edges from G so that the total i-th weights of all vertices in each component is at least li and at most ui for each index i, 1 ≤ i ≤ q. The problem of findin...

This paper investigates partitions which have neither parts nor hook lengths divisible by $$p$$ , referred to as -core $$p'$$ -partitions. We show that the largest -partition corresponds longest walk on a graph with vertices $$\{0, 1, \ldots p-1\}$$ and labelled edges defined via addition modulo . also exhibit an explicit family of large -partitions, giving lower bound size such partition is sa...

A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The tree-partition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ ...

A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| 6 1. The minimum cardinality of such a dominating set is denoted by γe(G) and is called equitable domination number of G. In this paper we introduce the equitable edge domination and equitable edge domatic number in a graph, exact...

For a nonempty graph G = (V, E), a signed edge-domination of G is a function f : E(G) → {1,−1} such that ∑e′∈NG [e] f (e′) ≥ 1 for each e ∈ E(G). The signed edge-domatic number of G is the largest integer d for which there is a set { f1, f2, . . . , fd} of signed edge-dominations of G such that ∑d i=1 fi (e) ≤ 1 for every e ∈ E(G). This paper gives an original study on this concept and determin...