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Xingxing Yu

Publications and source records attributed to Xingxing Yu.

40 records · Page 3Linked to original sources

arXiv2016

The Kelmans-Seymour conjecture III: 3-vertices in $K_4^-$

Let $G$ be a 5-connected nonplanar graph and let $x_1,x_2,y_1,y_2\in V(G)$ be distinct, such that $G[\{x_1,x_2,y_1,y_2\}]\cong K_4^-$ and $y_1y_2\notin E(G)$. We show that one of the following holds: $G-x_1$ contains $K_4^-$, or $G$ contains a $K_4^-$ in which $x_1$ is of degree 2, or $G$ contains a $TK_5$ in which $x_1$ is not a branch vertex, or $\{x_2,y_1,y_2\}$ may be chosen so that for any distinct $z_0, z_1\in N(x_1)-\{x_2,y_1,y_2\}$, $G-\{x_1v:v\notin \{z_0, z_1,x_2, y_1,y_2\}\}$ contains $TK_5$. This result will be used to prove the Kelmans-Seymour conjecture.

math.CO↗

arXiv2016

Induced Forests in Bipartite Planar Graphs

Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.

math.CO↗

arXiv2016

The Kelmans-Seymour conjecture II: 2-vertices in $K_4^-$

We use $K_4^-$ to denote the graph obtained from $K_4$ by removing an edge, and use $TK_5$ to denote a subdivision of $K_5$. Let $G$ be a 5-connected nonplanar graph and $\{x_1,x_2,y_1,y_2\}\subseteq V(G)$ such that $G[\{x_1,x_2,$ $y_1,y_2\}]\cong K_4^-$ with $y_1y_2\notin E(G)$. Let $w_1,w_2,w_3\in N(y_2)-\{x_1,x_2\}$ be distinct. We show that $G$ contains a $TK_5$ in which $y_2$ is not a branch vertex, or $G-y_2$ contains $K_4^-$, or $G$ has a special 5-separation, or $G-\{y_2v:v\notin \{w_1,w_2,w_3,x_1,x_2\}\}$ contains $TK_5$.

math.CO↗

arXiv2015

The Kelmans-Seymour conjecture I: special separations

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^-$, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture.

math.CO↗