arXiv subjects

Diogo Kendy Matsumoto

Publications and source records attributed to Diogo Kendy Matsumoto.

5 recordsLinked to original sources

arXiv2026

Characterization and chromatic number of triangle-free graphs with diameter 2

In this paper, we consider triangle-free graphs with diameter 2. If a triangle-free graph $G$ with diameter 2 is not isomorphic to a star, then the radius of $G$ is also 2, where such a graph is also called a $2$-self-centered graph. Shekarriz et al. [A characterization for 2-self-centered graphs, Discuss. Math. Graph Theory 38 (2018), 27--37.] gave a characterization of 2-self-centered graphs. However, there is a slight flaw in their characterization. Thus, in this paper, we modify it and prove an accurate characterization of those graphs. Furthermore, by using our characterization, we prove some results concerning the chromatic number of triangle-free graphs with diameter 2.

math.CO↗

arXiv2024

Travel groupoids on complete multipartite graphs

A travel groupoid is an algebraic system satisfying two suitable conditions, which has a relation to graphs. In this article, we characterize travel groupoids on finite complete multipartite graphs, and we give the numbers of travel groupoids on the complete multipartite graphs.

math.CO↗

arXiv2017

Quiver-theoretical approach to dynamical Yang-Baxter maps

A dynamical Yang-Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang-Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang-Baxter maps. Our key observation is that the category of dynamical sets over a set $Λ$, introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang-Baxter maps, can be embedded into the category of quivers with vertices $Λ$. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang-Baxter maps and extend his construction to obtain a new class of dynamical Yang-Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang-Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.

math.QA↗

arXiv2016

A construction of smooth travel groupoids on finite graphs

A travel groupoid is an algebraic system related with graphs. In this paper, we give an algorithm to construct smooth travel groupoids for any finite graph. This algorithm gives an answer of L.~Nebesk$\acute{\mbox{y}}$'s question, "Does there exists a connected graph $G$ such that $G$ has no smooth travel groupoid?", in finite cases.

math.CO↗

arXiv2011

Combinatorial aspects of dynamical Yang-Baxter maps and dynamical braces

In this article we propose an algebraic system, which is an abelian group $(A,+)$ with a family of non-associative and non-(left)distributive multiplications $\{\cdot_λ\}_{λ\in H}$. We call this algebraic system dynamical brace. The dynamical brace corresponds to a certain dynamical Yang-Baxter map (which is left nondegenerate and satisfy the unitary condition). Combinatorial aspects of the dynamical brace give us a correspondence between the dynamical brace and a certain family of subsets of $A\rtimes Aut(A)$. From this viewpoint we give an interpretation and examples of the dynamical brace.

math.RA↗