arXiv subjects

Ringi Kim

Publications and source records attributed to Ringi Kim.

22 records · Page 2Linked to original sources

arXiv2016

Cycles with two blocks in $k$-chromatic digraphs

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $c(k,\ell)$ is an oriented cycle which consists of two internally (vertex) disjoint directed paths of lengths at least $k$ and $\ell$, respectively, from a vertex to another one. A problem of Addario-Berry, Havet and Thomassé (2007) asked if, given positive integers $k$ and $\ell$ such that $k+\ell\ge 4$, any strongly connected digraph $D$ containing no $c(k,\ell)$ has chromatic number at most $k+\ell-1$. In this paper, we show that such digraph $D$ has chromatic number at most $O((k+\ell)^2)$, improving the previous upper bound $O((k+\ell)^4)$ obtained by Cohen, Havet, Lochet and Nisse (2016). In fact, we are able to find a digraph which shows that the answer to the above problem is no. We also show that if in addition $D$ is Hamiltonian, then its underlying simple graph is $(k+\ell-1)$-degenerate and thus the chromatic number of $D$ is at most $k+\ell$, which is tight.

math.CO↗

arXiv2015

Unavoidable induced subgraphs in large graphs with no homogeneous sets

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.

math.CO↗

arXiv2014

Disjoint dijoins

A dijoin in a digraph is a set of edges meeting every directed cut. D. R. Woodall conjectured in 1976 that if G is a digraph, and every directed cut of G has at least k edges, then there are k pairwise disjoint dijoins. This remains open, but a capacitated version is known to be false. In particular, A. Schrijver gave a digraph G and a subset S of its edge-set, such that every directed cut contains at least two edges in S, and yet there do not exist two disjoint dijoins included in S. In Schrijver's example, G is planar, and the subdigraph formed by the edges in S consists of three disjoint paths. We conjecture that when k = 2, the disconnectedness of S is crucial: more precisely, that if G is a digraph, and S is a subset of the edges of G that forms a connected subdigraph (as an undirected graph), and every directed cut of G contains at least two edges in S, then we can partition S into two dijoins. We prove this in two special cases: when G is planar, and when the subdigraph formed by the edges in S is a subdivision of a caterpillar.

math.CO↗

arXiv2013

The Minimal Automorphism-Free Tree

A finite tree $T$ with $|V(T)| \geq 2$ is called {\it automorphism-free} if there is no non-trivial automorphism of $T$. Let $\mathcal{AFT}$ be the poset with the element set of all finite automorphism-free trees (up to graph isomorphism) ordered by $T_1 \preceq T_2$ if $T_1$ can be obtained from $T_2$ by successively deleting one leaf at a time in such a way that each intermediate tree is also automorphism-free. In this paper, we prove that $\mathcal{AFT}$ has a unique minimal element. This result gives an affirmative answer to the question asked by Rupinski.

math.CO↗