arXiv subjects
Publications and source records attributed to Dang Dinh Hanh.
A \textit{cut-edge} of a connected graph is an edge whose deletion increases the number of components. In this paper, we first state some conditions for a planar graph to have a few cut-edges. After that, we use the main results to study the conditions for a colored planar graph to have a bounded conflict-free connection number.
In this article, we will prove that if $G$ is a connected claw-free graph and either $σ_6(G)\geq |G|-5$ or $σ_7(G)\geq |G|-2$, here $σ_k(G)$ is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a spanning tree with at most two branch vertices.
In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with $σ_4(G)\geq n-1$ contains a spanning tree with at most $3$ leaves. In this paper, we prove an analogue of Kyaw's result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with $σ_5(G)\geq n-1$ contains a spanning tree with at most $4$ leaves. Moreover, the degree sum condition `$σ_5(G)\geq n-1$' is best possible.
Let $T$ be a tree, a vertex of degree one is called a leaf. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this note, we give a sharp sufficient condition to show that a $K_{1,t}-$free graph has a spanning tree whose stem has a few leaves. By applying the main result, we give improvements of previous related results.
A braided $Ann$-category $\mathcal A$ is an $Ann$-category $\mathcal A$ together with a braiding $c$ such that $(\mathcal A, \otimes, a, c, (1,l,r))$ is a braided tensor category, moreover $c$ is compatible with the distributivity constraints. According to the structure transport theorem, the paper shows that each braided $Ann$-category is equivalent to a braided $Ann$-category of the type $(R,M)$, hence the proof of the classification theorem for braided $Ann$-categories by the cohomology of commutative rings is presented.
Dual monoidal category $\mathcal C^\ast$ of a monoidal functor $F:\mathcal C\to \mathcal V$ has been constructed by S. Majid. In this paper, we extend the construction of dual structures for an Ann-functor $F:\mathcal B\to \mathcal A$. In particular, when $F=id_{\mathcal A}$, then the dual category $\mathcal A^{\ast}$ is indeed the center of $\mathcal A$ and this is a braided Ann-category.
Regular Ann-functor classification problem has been solved with Shukla cohomology. In this paper, we would like to present a solution to the above problem in the general case and in the case of strong Ann-functors with, respectively, Mac Lane cohomology and Hochschild cohomology.