arXiv subjects
Publications and source records attributed to Gerard Jennhwa Chang.
A flat torus is the quotient of the Euclidean plane over a lattice generated by a basis, and an axis-aligned rectangular tiling of a flat torus is a partition into finitely many rectangles whose sides are axis-aligned. We provide the minimum sum of the perimeter of rectangles for an axis-aligned rectangular tiling, and prove that it is attainable by either exactly one rectangle or exactly two rectangles.
For a graph $G$ in which vertices are either black or white, a zero forcing process is an iterative vertex color changing process such that the only white neighbor of a black vertex becomes black in the next time step. A zero forcing set is an initial subset of black vertices in a zero forcing process ultimately expands to include all vertices of the graph; otherwise we call its complement a zero blocking set. The zero blocking number $B(G)$ of $G$ is the minimum size of a zero blocking set. This paper determines zero blocking numbers of the union and the join of two graphs. It also determines all minimum zero blocking sets of hypercubes. Finally, a linear-time algorithm for the zero blocking numbers of trees is given.
In a zero forcing process, vertices of a graph are colored black and white initially, and if there exists a black vertex adjacent to exactly one white vertex, then the white vertex is forced to be black. A zero blocking set is an initial set of white vertices in a zero forcing process such that ultimately there exists a white vertex. The zero blocking number is the minimum size of a zero blocking set. This paper gives the exact value of the zero blocking number of grid graphs.
This paper extends the results by Afzali, Ghodrati and Maimani for the zero blocking numbers of Kneser graphs and Johnson graphs to generalized Kneser graphs and generalized Johnson graphs.
A strong $k$-edge-coloring of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index $χ'_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. Denote $σ(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$. It is easy to see that $σ(G) \le χ'_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $Δ$, it was proved that $χ'_s(G) \le 4 Δ+4$ by using the Four Color Theorem. The upper bound was then reduced to $4Δ$, $3Δ+5$, $3Δ+1$, $3Δ$, $2Δ-1$ under different conditions for $Δ$ and the girth. In this paper, we prove that if the girth of a planar graph $G$ is large enough and $σ(G)\geq Δ(G)+2$, then the strong chromatic index of $G$ is precisely $σ(G)$. This result reflects the intuition that a planar graph with a large girth locally looks like a tree.
Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S) \ge 2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.