A contribution is made to the classification of lattice-like total perfect codes in integer lattices Λn via pairs (G, Φ) formed by abelian groups G and homomorphisms Φ: Zn → G. A conjecture is posed that the cited contribution covers all possible cases. A related conjecture on the unfinished work on open problems on lattice-like perfect dominating sets in Λn with induced components that are parallel paths of length > 1 is posed as well.

The looseness of a triangulation G on a closed surface F2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have [...] and this bound is sharp. For a triangulation G on a non-spherical surface F2, we have...

>We prove that the domination number γ(T) of a tree T on n ≥ 3 vertices and with n₁ endvertices satisfies inequality γ(T) ≥ (n+2-n₁)/3 and we characterize the extremal graphs.

In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

The open neighborhood $N_G\left(e\right)$ of an edge $e$ in a graph $G$ is the set consisting of all edges having a common end-vertex with $e$. Let $f$ be a function on $E\left(G\right)$, the edge set of $G$, into the set $\{-1,1\}$. If ${\sum }_{x\in N_G\left(e\right)}f\left(x\right)\ge 1$ for each $e\in E\left(G\right)$, then $f$ is called a signed edge total dominating function of $G$. The minimum of the values ${\sum }_{e\in E\left(G\right)}f\left(e\right)$, taken over all signed edge total dominating function $f$ of $G$, is called the signed edge total domination number of $G$ and is denoted by ${\gamma }_{st}^{\text{'}}\left(G\right)$. Obviously, ${\gamma }_{st}^{\text{'}}\left(G\right)$ is defined only for graphs $G$ which have no connected components...