arXiv subjects

Tayo Charles Adefokun

Publications and source records attributed to Tayo Charles Adefokun.

4 recordsLinked to original sources

arXiv2025

Hamiltonian Complete Number of Some Variants of Caterpillar Graphs

A graph $G$ is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are adjacent. For a non-Hamiltonian graph $G$, the Hamiltonian complete number $λ_H(G)$ is the minimum number of edges that must be added to $G$ to make it Hamiltonian. We focus on both regular and irregular caterpillar graphs, deriving explicit formulas for $λ_H(G)$ in various cases. Specifically, we show that for a regular caterpillar graph $G_{n(k)}$ where each vertex on the central path is adjacent to $k$ leaves, $λ_H(G_{n(k)}) = n(k-1)$. We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for $λ_H(G)$ in these cases. Our results contribute to the understanding of Hamiltonian properties in tree-like structures and have potential applications in network design and optimization.

math.CO↗

arXiv2019

On Radio Number of Stacked-Book Graphs

A Stacked-book graph $G_{m,n}$ results from the Cartesian product of a star graph $S_m$ and path $P_n$, where $m$ and $n$ are the orders of $S_m$ and $P_n$ respectively. A radio labeling problem of a simple and connected graph, $G$, involves a non-negative integer function $f:V(G)\rightarrow \mathbb Z^+$ on the vertex set $V(G)$ of G, such that for all $u,v \in V(G)$, $|f(u)-f(v)| \geq \textmd{diam}(G)+1-d(u,v)$, where $\textmd {diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the shortest distance between $u$ and $v$. Suppose that $f_{min}$ and $f_{max}$ are the respective least and largest values of $f$ on $V(G)$, then, span$f$, the absolute difference of $f_{min}$ and $f_{max}$, is the span of $f$ while the radio number $rn(G)$ of $G$ is the least value of span$f$ over all the possible radio labels on $V(G)$. In this paper, we obtain the radio number for the stacked-book graph $G_{m,n}$ where $m \geq 4$ and $n$ is even, and obtain bounds for $m=3$ which improves existing upper and lower bounds for $G_{m,n}$ where $m=3$.

math.CO↗

arXiv2017

Some bounds on the maximum induced matching numbers of certain grids

An induced matching $M$ in a graph $G$ is a matching in $G$ that is also the edge set of an induced subgraph of $G$. That is, any edge not in $M$ must have no more than one incident vertex saturated by $M$. The maximum size $|M|$ of an induced matching $M$ of $G$ is maximum induced matching number of $G$, which is denoted by $\textrm{Max}(G)$. In this article, we obtain upper bounds for $\textrm{Max}(G)$, for $G=G_{n,m}$, grids with $n,m \geq 9$, $m\equiv 1 \mod 4$ and $nm$ odd.

math.CO↗

arXiv2015

$L(1,1)-$ Labeling of Direct Product of Cycles

An $L(1,1)$-labeling of a graph $G$ is an assignment of labels from $\{0,1 \cdots, k \}$ to the vertices of $G$ such that two vertices that are adjacent or have a common neighbor receive distinct labels. The $λ_1^1-$ number, $λ_1^1(G)$ of $G$ is the minimum value $k$ such that $G$ admits an $L(1,1)$ labeling. We establish the $λ_1^1-$ numbers for direct product of cycles $C_m \times C_n$ for all positive $m, n \geq 3$, where both $m,n$ are even or when one of them is even and the other odd.

math.CO↗