Let $G = (V, E)$ be a finite simple undirected graph without $K_2$
components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a
local antimagic labeling if for any two adjacent vertices $u$ and $v$, they
have different vertex sums, i.e., $w(u) \neq w(v)$, where the vertex sum $w(u)
= \sum_{e \in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus
any local antimagic labeling induces a proper vertex coloring of $G$ where the
vertex $v$ is assigned the color (vertex sum) $w(v)$. The local antimagic
chromatic number $\chi_{la}(G)$ is the minimum number of colors taken over all
colorings induced by local antimagic labelings of $G$. It was conjectured
\cite{Aru-Wang} that for every tree $T$ the local antimagic chromatic number
$l+ 1 \leq \chi_{la} ( T )\leq l+2$, where $l$ is the number of leaves of $T$.
In this article we verify the above conjecture for complete full $t$-ary trees,
for $t \geq 2$. A complete full $t$-ary tree is a rooted tree in which all
nodes have exactly $t$ children except leaves and every leaf is of the same
depth. In particular we obtain that the exact value for the local antimagic
chromatic number of all complete full $t$-ary trees is $ l+1$ for odd $t$.
"There and Back Again" (TABA) is a programming pattern where the recursive
calls traverse one data structure and the subsequent returns traverse another.
This article presents new TABA examples, refines existing ones, and formalizes
both their control flow and their data flow using the Coq Proof Assistant. Each
formalization mechanizes a pen-and-paper proof, thus making it easier to "get"
TABA. In addition, this article identifies and illustrates a tail-recursive
variant of TABA, There and Forth Again (TAFA) that does not come back but goes
forth instead with more tail calls.
Domination-type parameters are difficult to manage in Cartesian product
graphs and there is usually no general relationship between the parameter in
both factors and in the product graph. This is the situation of the domination
number, the Roman domination number or the $2$-domination number, among others.
Contrary to what happens with the domination number and the Roman domination
number, the $2$-domination number remains unknown in cylinders, that is, the
Cartesian product of a cycle and a path and in this paper, we will compute this
parameter in the cylinders with small cycles. We will develop two algorithms
involving the $(\min,+)$ matrix product that will allow us to compute the
desired values of $\gamma_2(C_n\Box P_m)$, with $3\leq n\leq 15$ and $m\geq 2$.
We will also pose a conjecture about the general formulae for the
$2$-domination number in this graph class.