arXiv subjects

Long-Tu Yuan

Publications and source records attributed to Long-Tu Yuan.

25 records · Page 2Linked to original sources

arXiv2020

Extremal graphs of the $k$-th power of paths

An extremal graph for a given graph $H$ is a graph with maximum number of edges on fixed number of vertices without containing a copy of $H$. The $k$-th power of a path is a graph obtained from a path and joining all pair of vertices of the path with distance less than $k$. Applying a deep theorem of Simonovits, we characterize the extremal graphs of the $k$-th power of paths. This settles a conjecture posed by Xiao, Katona, Xiao and Zamora in a stronger form.

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arXiv2020

On the anti-Ramsey numbers of linear forests

For a fixed graph $F$, the $\textit{anti-Ramsey number}$, $AR(n,F)$, is the maximum number of colors in an edge-coloring of $K_n$ which does not contain a rainbow copy of $F$. In this paper, we determine the exact value of anti-Ramsey numbers of linear forests for sufficiently large $n$, and show the extremal edge-colored graphs. This answers a question of Fang, Győri, Lu and Xiao.

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arXiv2019

Anti-Ramsey numbers of graphs with some decomposition family sequences

For a given graph $H$, the anti-Ramsey number of $H$ is the maximum number of colors in an edge-coloring of a complete graph which does not contain a rainbow copy of $H$. In this paper, we extend the decomposition family of graphs to the decomposition family sequence of graphs and show that $K_5$ is determined by its decomposition family sequence. Based on this new graph notation, we determine the anti-Ramsey numbers for new families of graphs, including the Petersen graph, vertex-disjoint union of cliques, etc., and characterize the extremal colorings.

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arXiv2017

A Variation of the Erdős-Sós Conjecture in Bipartite Graphs

The Erdős-Sós Conjecture states that every graph with average degree more than $k-2$ contains all trees of order $k$ as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an $(n,m)$-bipartite graph which does not contain all $(k,l)$-bipartite trees for given integers $n\ge m$ and $k\ge l$. In particular, we determine that the maximum size of an $(n,m)$-bipartite graph which does not contain all $(n,m)$-bipartite trees as subgraphs (or all $(k,2)$-bipartite trees as subgraphs, respectively). Furthermore, all these extremal graphs are characterized.

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arXiv2016

Turán numbers for disjoint paths

The Turán number of a graph $H$, $ex(n,H)$, is the maximum number of edges in any graph of order $n$ which does not contain $H$ as a subgraph. Lidický, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large and proved that the extremal graph is unique, where $F_m$ is disjoint paths of $P_{k_1}, \ldots, P_{k_m}$ [Lidický,B., Liu,H. and Palmer,C. (2013). On the Turán number of forests. Electron. J. Combin. 20(2) Paper 62, 13 pp]. In this paper, by mean of a different approach, we determine $ex(n, F_m)$ for all integers $n$ with minor conditions, which extends their partial results. Furthermore, we partly confirm the conjecture proposed by Bushaw and Kettle for $ex(n, k\cdot P_l)$ [Bushaw,N. and Kttle,N. (2011) Turán numbers of multiple paths and equibipartite forests. Combin. Probab. Comput. 20 837-853]. Moreover, we show that there exist two family graphs $F_m$ and $F_m^{\prime}$ such that $ex(n, F_m)=ex(n, F_m^{\prime})$ for all integers $n$, which is related to an old problem of Erdős and Simonovits.

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arXiv2015

The Turan Number of Disjoint Copies of Paths

The Turán number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we determine the value $ex(n, k\cdot P_3)$ and characterize all extremal graphs. This extends a result of Bushaw and Kettle [N. Bushaw and N. Kettle, Turán Numbers of multiple and equibipartite forests, Combin. Probab. Comput., 20(2011) 837-853.], which solved the conjecture proposed by Gorgol in [I. Gorgol. Turán numbers for disjoint copies of graphs. {\it Graphs Combin.}, 27 (2011) 661-667.].

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arXiv2014

On the Erdos-Sos Conjecture for Graphs on n=k+4 Vertices

The Erdős-Sós Conjecture states that if $G$ is a simple graph of order $n$ with average degree more than $k-2,$ then $G$ contains every tree of order $k$. In this paper, we prove that Erdős-Sós Conjecture is true for $n=k+4$.

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