Tag: 김진하

An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or $\Delta\ge 6$, every connected $n$-vertex graph of maximum degree at most $\Delta$ has an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\frac{\Delta}{ \lfloor\Delta^2/4\rfloor+\Delta })n$.
This is joint work with Eun-Kyung Cho, Minki Kim, and Sang-il Oum.

Given a graph G=(V,E), the independence complex of G is the abstract simplicial complex I(G) on V whose faces are the independent sets of G. A graph is ternary if it does not contain an induced cycle of length divisible by three. Kalai and Meshulam conjectured that if G is ternary then the sum of the Betti numbers of I(G) is either 0 or 1. In this talk, I will introduce a result by Zhang and Wu, which proves the Kalai-Meshulam conjecture.

Let $G$ be a graph on the vertex set $V$. A vertex subset $W \subset V$ is a cover of $G$ if $V \setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex on $V$ whose faces are non-covers of $G$. Then the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. In this talk, I will show the $(|V(G)|-i\gamma(G)-1)$-collapsibility of the non-cover complex of a graph $G$ where $i\gamma(G)$ denotes the independence domination number of $G$ using the minimal exclusion sequence method. This is joint work with Ilkyoo Choi and Boram Park.