Iterated colorings of graphs

AbstractFor a graph property P, in particular maximal independence, minimal domination and maximal irredundance, we introduce iterated P-colorings of graphs. The six graph parameters arising from either maximizing or minimizing the number of colors used for each property, are related by an inequality chain, and in this paper we initiate the study of these parameters. We relate them to other well-studied parameters like chromatic number, give alternative characterizations, find graph classes where they differ by an arbitrary amount, investigate their monotonicity properties, and look at algorithmic issues

Transitivity on subclasses of chordal graphs

Let $G=(V, E)$ be a graph, where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive partition of a given graph with the maximum number of partitions. It was known that the decision version of \textsc{Maximum Transitivity Problem} is NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete Mathematics}, 278, 2004]. In this paper, we first prove that this problem can be solved in linear time for \emph{split graphs} and for the \emph{complement of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss Nordhaus-Gaddum type relations for transitivity and provide counterexamples for an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018]. Finally, we characterize transitively critical graphs having fixed transitivity.Comment: arXiv admin note: text overlap with arXiv:2204.1314

A Study Of The Upper Domatic Number Of A Graph

Given a graph G we can partition the vertices of G in to k disjoint sets. We say a set A of vertices dominates another set of vertices, B, if for every vertex in B there is some adjacent vertex in A. The upper domatic number of a graph G is written as D(G) and defined as the maximum integer k such that G can be partitioned into k sets where for every pair of sets A and B either A dominates B or B dominates A or both. In this thesis we introduce the upper domatic number of a graph and provide various results on the properties of the upper domatic number, notably that D(G) is less than or equal to the maximum degree of G, as well as relating it to other well-studied graph properties such as the achromatic, pseudoachromatic, and transitive numbers