The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties...

The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = ∑uv∈E(G) |dG(u) − dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and sym- metric difference. We give exact expressions or (sharp) upper bounds on the irregularity of graphs under the above mentioned...

We give explicit non-recursive formulas to compute the Josephus-numbers $j(n,2,i)$ and $j(n,3,i)$ and explicit upper and lower bounds for $j(n,k,i)$ (where $k\ge 4$) which differ by $2k-2$ (for $k=4$ the bounds are even better). Furthermore we present a new fast algorithm to calculate $j(n,k,i)$ which is based upon the mentioned bounds.

Let $k$ be a positive integer, and let $G$ be a simple graph with vertex set $V\left(G\right)$. A $k$-dominating set of the graph $G$ is a subset $D$ of $V\left(G\right)$ such that every vertex of $V\left(G\right)-D$ is adjacent to at least $k$ vertices in $D$. A $k$-domatic partition of $G$ is a partition of $V\left(G\right)$ into $k$-dominating sets. The maximum number of dominating sets in a $k$-domatic partition of $G$ is called the $k$-domatic number$d_k\left(G\right)$. In this paper, we present upper and lower bounds for the $k$-domatic number, and we establish Nordhaus-Gaddum-type results. Some of...

For a nontrivial connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u-v$ path in $G$. Detour distance is a metric on the vertex set of $G$. For each integer $k$ with $1\le k\le n-1$, a coloring $c:V\left(G\right)\to \mathbb{N}$ is a $k$-metric coloring of $G$ if $\left|c\right(u)-c(v\left)\right|+D(u,v)\ge k+1$ for every two distinct vertices $u$ and $v$ of $G$. The value ${\chi }_m^k\left(c\right)$ of a $k$-metric coloring $c$ is the maximum color assigned by $c$ to a vertex of $G$ and the $k$-metric chromatic number ${\chi }_m^k\left(G\right)$ of $G$ is the minimum value of a $k$-metric coloring of $G$. For every...

Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V with f(v) = Ø the condition ∪u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. The weight of a kRDF f is the value w(f) = ∑v∈V |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set 1,2, ...,k such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u ∈ N(v)f(u) = 1,2, ...,k is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set $f₁,f₂,...,f_d$ of k-rainbow dominating functions on G with the property that ${\sum }_{i=1}^d\left|f_i\left(v\right)\right|\le k$ for each v ∈ V(G), is called a k-rainbow dominating family (of...