This paper investigates the dynamic interplay between Artificial Intelligence (AI) and human logic in the domain of mathematical problem-solving. By critically examining a series of case studies, we compare the efficacy of AI-generated solutions, particularly those offered by ChatGPT, against traditional human problem-solving methods. The study employs various mathematical challenges, ranging from abstract logical puzzles to applied numerical problems, to evaluate AI's problem-solving approach and alignment with human cognitive processes. Our analysis highlights instances where AI's computational strategies complement or diverge from human reasoning, shedding light on AI's potential and limitations in deciphering mathematical problems. Furthermore, we explore the implications of integrating AI tools in educational contexts, specifically their role in enhancing students' mathematical problem-solving skills. The paper aims to contribute to the ongoing discourse on the optimal utilization of AI in education, proposing a balanced approach that leverages AI's computational power while fostering the depth and creativity of human logic. Through this comparative study, we advocate for a collaborative model where AI and human reasoning merge to enrich the educational landscape, particularly in the teaching and learning of mathematics.

The domination (number) of a graph $G=(V,E)$, denoted by $\gamma \left(G\right)$, is the size of the minimum dominating sets of $V\left(G\right)$, also known as $\gamma$-sets. As such, the dominion of G, denoted by $\zeta \left(G\right)$, counts all its $\gamma$-sets. We proved a conjecture from one of the authors on the dominion of cycles $C_{3k-1}$ and $C_{3k-2}$, $k\ge 2$. Further, we found the formulae and recurrence relations for the dominions of several grids, $G_{m,n}$, with $2\le m\le 4$ and other results when $m\le 9$ and $n\le 20$. In general, domination and dominion play important roles in assessing certain vulnerabilities of any given network system.

We prove that chordal graphs are chromatic-choosable and present a decomposition theorem to help estimate the choice numbers for certain classes of chordless and non-chordal graphs.