arXiv subjects

Ruonan Li

Publications and source records attributed to Ruonan Li.

21 records · Page 2Linked to original sources

arXiv2017

Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.

math.CO↗

arXiv2017

Decomposing edge-colored graphs under color degree constraints

For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then $V(G)$ can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least $2$. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.

math.CO↗

arXiv2016

Cyclability of $id$-cycles in graphs

Let $G$ be a graph on $n$ vertices and $C'=v_0v_1\cdots v_{p-1}v_0$ a vertex sequence of $G$ with $p\geq 3$ ($v_i\neq v_j$ for all $i,j=0,1,\ldots,p-1$, $i\neq j$). If for any successive vertices $v_i$, $v_{i+1}$ on $C'$, either $v_iv_{i+1}\in E(G)$ or both of the first implicit-degrees of $v_i$ and $v_{i+1}$ are at least $n/2$ (indices are taken modulo $p$), then $C'$ is called an $id$-cycle of $G$. In this paper, we prove that for every $id$-cycle $C'$, there exists a cycle $C$ in $G$ with $V(C')\subseteq V(C)$. This generalizes several early results on the Hamiltonicity and cyclability of graphs.

math.CO↗