arXiv subjects
Publications and source records attributed to Long-Tu Yuan.
We determine the generalized Tur\'an number of vertex-disjoint paths of length two and characterize all corresponding extremal graphs. Our proof combines the Lov\'asz form of the Kruskal--Katona theorem with a discrete convexity argument.
For a graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with $\mathrm{ex}(n,F)+q$ edges, where $\mathrm{ex}(n,F)$ is the maximum number of edges in an $n$-vertex $F$-free graph. Let $c(n,F)$ be the minimum number of copies obtained by adding one edge to an extremal $F$-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that $h_F(n,q)\ge q\,c(n,F)$. Ma and Yuan recently constructed stable graph counterexamples for every fixed $q\ge4$; they asked whether the one-edge equality $h_F(n,1)=c(n,F)$ might still hold for every graph $F$ containing a cycle. We give a negative answer to their question. For each integer $t\ge6$, let $H_t$ be obtained from the $t$-vertex path by replacing each edge with a $3t$-page book, using disjoint page vertices for different path edges. Then $h_{H_t}(n,1)<c(n,H_t)$ for infinitely many values of $n$. Moreover, by taking $t$ large, the ratio $h_{H_t}(n,1)/c(n,H_t)$ can be made arbitrarily small along infinitely many values of $n$.
The expansion $F^{\triangle}$ of a graph $F$ is the graph obtained from $F$ by replacing each edge with a triangle. Lv \etal proposed a conjecture on the maximum number of triangles in a graph without $P_k^{\triangle}$ or $C_k^{\triangle}$ for every $k \ge 4$. Their conjecture was confirmed in previous work for $P_k^{\triangle}$ when $k \ge 4$ and $C_k^{\triangle}$ when $k \ge 5$. In this note, we resolve the remaining case $C_4^{\triangle}$, demonstrating that this is the only counterexample to their conjecture.
The Turán number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Turán numbers for a path of arbitrary length. In 1975, Faudree and Schelp, and independently in 1977, Kopylov determined the exact values of Turán numbers of paths with arbitrary length. In this paper, we determine the Turán number of two vertex-disjoint paths of odd order at least 4. Together with previous works, we determine the exact Turán numbers of two vertex-disjoint paths completely. This confirms the first $k=2$ case of a conjecture proposed by Yuan and Zhang in 2021, which generalizes the Turán number formula of paths due to Faudree-Schelp, and Kopylov in a broader setting. Our main tools include a refinement of Pósa's rotation lemma, a stability result of Kopylov's theorem on cycles, and a recent inequality on circumference, minimum degree, and clique number of a 2-connected graph.
The bipartite independence number of a graph $G$, denoted by $\widetildeα(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s + t = q + 1$, such that for any two disjoint subsets $A, B \subseteq V(G)$ with $|A| = s$ and $|B| = t$, there exists an edge between $A$ and $B$. In this paper, we prove that for a 2-connected graph $G$ of order at least three, if $\max\{d_G(x), d_G(y)\} \ge \widetildeα(G)$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian. Moreover, we prove that if $G$ is 3-connected and $\max\{d_G(x), d_G(y)\} \ge \widetildeα(G)+1$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian-connected. Our results generalize the recent work by Li and Liu.
The problem of determining the maximum number of copies of $T$ in an $H$-free graph, for any graphs $T$ and $H$, was considered by Alon and Shikhelman. This is a variant of Turán's classical extremal problem. We show lower and upper bounds for the maximum number of $s$-cliques in a graph with no disjoint copies of arbitrary graph. We also determine the maximum number of $s$-cliques in an $n$-vertex graph that does not contain a disjoint union of $k$ paths of length two when $k=2,3$, or $s\geqslant k+2$, or $n$ is sufficiently large, this partly confirms a conjecture posed by Chen, Yang, Yuan, and Zhang \cite{2024Chen113974}.
The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained from $F$ by adding a new vertex to each edge such that different edges receive different vertices. For large $n$, we establish tight upper bounds for: The maximum number of edges in an $n$-vertex $3$-graph that does not contain $T^3$ for certain class $\mathcal{T}$ of trees, sharpening (partially) a result of Kostochka--Mubayi--Verstraëte. The minimum number of colors needed to color the complete $n$-vertex $3$-graph to ensure the existence of a rainbow copy of $F^3$ when $F$ is a graph obtained from some tree $T\in \mathcal{T}$ by adding a new edge, extending anti-Ramsey results on $P_{2t}^3$ by Gu--Li--Shi and $C_{2t}^3$ by Tang--Li--Yan. The maximum number of edges in an $n$-vertex $3$-graph whose shadow does not contain the shadow of $C_{k}^3$ or $T^3$ for $T\in \mathcal{T}$, answering a question of Lv \etal on generalized Turán problems.
We determine the maximum number of a graph without containing the 2-power of a Hamilton path. Using this result, we establish a spectral condition for a graph containing the 2-power of a Hamilton path.
The supersaturation problem for a given graph $F$ asks for the minimum number $h_F(n,q)$ of copies of $F$ in an $n$-vertex graph with $ex(n,F)+q$ edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of $q$ (which is linear in $n$) for cliques $F$ such that $h_F(n,q)$ equals the minimum number $t_F(n,q)$ of copies of $F$ obtained from a maximum $F$-free $n$-vertex graph by adding $q$ new edges. A breakthrough result of Mubayi extends this line of research from cliques to color-critical graphs $F$, and this was further strengthened by Pikhurko and Yilma who established the equality $h_F(n,q)=t_F(n,q)$ for $1\leq q\leq ε_F n$ and sufficiently large $n$. In this paper, we present several results on the supersaturation problem that extend beyond the existing framework. Firstly, we explicitly construct infinitely many graphs $F$ with restricted properties for which $h_F(n,q)<q\cdot t_F(n,1)$ holds when $n\gg q\geq 4$, thus refuting a conjecture of Mubayi. Secondly, we extend the result of Pikhurko-Yilma by showing the equality $h_F(n,q)=t_F(n,q)$ in the range $1\leq q\leq ε_F n$ for any member $F$ in a diverse and abundant graph family (which includes color-critical graphs, disjoint unions of cliques $K_r$, and the Petersen graph). Lastly, we prove the existence of a graph $F$ for any positive integer $s$ such that $h_F(n,q)=t_F(n,q)$ holds when $1\leq q\leq ε_F n^{1-1/s}$, and $h_F(n,q)<t_F(n,q)$ when $n^{1-1/s}/ε_F\leq q\leq ε_F n$, indicating that $q=Θ(n^{1-1/s})$ serves as the threshold for the equality $h_F(n,q)=t_F(n,q)$. We also discuss some additional remarks and related open problems.
For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a general approach for determining the maximum number of edges in an $n$-vertex $r$-graph that does not contain $t+1$ vertex-disjoint copies of $F$. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions. Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Simonovits~\cite{SI68} and Moon~\cite{Moon68} on complete graphs and can be viewed as a step towards a general density version of the classical Corrádi--Hajnal Theorem~\cite{CH63}.
The Erdős-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős-Simonovits type stability theorem in multi-partite graphs. Different from the Erdős-Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away to the extremal graphs for $H$. As an application, we solve a conjecture posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique
The edge blow-up of a graph is the graph obtained from replacing each edge of it by a clique of the same size where the new vertices of the cliques are all different. Wang, Hou, Liu and Ma determined the Turán number of the edge blow-up of trees except one particular case. Answering an problem posed by them, we determined the Turán number of this particular case.
The bipartite Turán number of a graph $H$, denoted by $ex(m,n; H)$, is the maximum number of edges in any bipartite graph $G=(X,Y; E)$ with $|X|=m$ and $|Y|=n$ which does not contain $H$ as a subgraph. In this paper, we determined $ex(m,n; F_{\ell})$ for arbitrary $\ell$ and appropriately large $n$ with comparing to $m$ and $\ell$, where $F_\ell$ is a linear forest which consists of $\ell$ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which does not contain $F_{\ell}$ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.
For a graph $H$, the Turán number of $H$, denoted by ex$(n,H)$, is the maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a $2$-edge-coloring of $K_n$. A wheel $W_m$ is a graph formed by connecting a single vertex to all vertices of a cycle of length $m-1$. The Turán number of $W_{2k}$ was determined by Simonovits in the 1960s. In this paper, we determine ex$(n,W_{2k+1})$ when $n$ is sufficiently large. We also show that, for sufficiently large $n$, $g(n,W_{2k+1})=\mbox{ex}(n,W_{2k+1})$ which confirms a conjecture posed by Keevash and Sudakov for odd wheels.
Given a graph $H$ and an integer $p$, the {\it edge blow-up} of $H$, denoted as $H^{p+1}$, is the graph obtained from replacing each edge in $H$ by a clique of size $p+1$ where the new vertices of the cliques are all different. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. In this paper, we determine the Turán numbers for edge blow-up of general graphs.
We determine the anti-Ramsey numbers for paths. This confirms a conjecture posed by Erdős, Simonovits and Sós in 1970s.
Combining Pósa's rotation lemma with a technique of Kopylov in a novel approach, we prove a generalization of the Erdős-Gallai theorems on cycles and paths. This implies a clique version of the Erdős-Gallai stability theorems and also provides alternative proofs for some recent results.
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.