arXiv subjects

Ruth Luo

Publications and source records attributed to Ruth Luo.

30 records · Page 2Linked to original sources

arXiv2020

Towards the Small Quasi-Kernel Conjecture

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally demicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs).

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arXiv2019

Super-pancyclic hypergraphs and bipartite graphs

We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with $|A| \geq 3$, $\mathcal H$ contains a Berge cycle with vertex set $A$. We mostly use the language of bipartite graphs, because every bipartite graph is the incidence graph of a multihypergraph. In particular, we extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Furthermore, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs.

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arXiv2019

On 2-connected hypergraphs with no long cycles

We give an upper bound for the maximum number of edges in an $n$-vertex 2-connected $r$-uniform hypergraph with no Berge cycle of length $k$ or greater, where $n\geq k \geq 4r\geq 12$. For $n$ large with respect to $r$ and $k$, this bound is sharp and is significantly stronger than the bound without restrictions on connectivity. It turned out that it is simpler to prove the bound for the broader class of Sperner families where the size of each set is at most $r$. For such families, our bound is sharp for all $n\geq k\geq r\geq 3$.

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arXiv2018

Avoiding long Berge cycles II, exact bounds for all $n$

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $k\geq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $k\geq r+4$.

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arXiv2018

On $r$-uniform hypergraphs with circumference less than $r$

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Győri, Katona and Lemons that for $n>r\geq k\geq 3$, every $n$-vertex $r$-uniform hypergraph with no Berge path of length $k$ has at most $\frac{(k-1)n}{r+1}$ edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least $2k$, and then translate the results into the language of multi-hypergraphs.

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arXiv2018

Avoiding long Berge cycles

Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then $\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$ divides $n-1$. We also show that the bound is attained only for connected $r$-uniform hypergraphs in which every block is the complete hypergraph $K^{(r)}_{k-1}$. We conjecture that our bound also holds in the case $k=r+2$, but the case of short cycles, $k\leq r+1$, is different.

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arXiv2018

A variation of a theorem by Pósa

A graph $G$ is $\ell$-hamiltonian if for any linear forest $F$ of $G$ with $\ell$ edges, $F$ can be extended to a hamiltonian cycle of $G$. We give a sharp upper bound for the maximum number of cliques of a fixed size in a non-$\ell$-hamiltonian graph. Furthermore, we prove stability for the bound: if a non-$\ell$-hamiltonian graph contains almost the maximum number of cliques, then it must be a subgraph of one of two examples.

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arXiv2017

The maximum number of cliques in graphs without long cycles

The Erdős--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of $s$-cliques in a graph with no path on $k$-vertices.

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arXiv2017

A forest building process on simple graphs

Consider the following process on a simple graph without isolated vertices: Order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a spanning forest of the graph. The probability of obtaining $k$ components in this process for complete bipartite graphs is determined as well as a formula for the expected number of components in any graph. A generic recurrence and some additional basic properties are discussed.

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arXiv2017

Stability in the Erdős--Gallai Theorem on cycles and paths, II

The Erdős--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erdős--Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e(G) \leq \max\{h(n,k,2),h(n,k,\lfloor \frac{k-1}{2}\rfloor)\}$, where $h(n,k,a) := {k - a \choose 2} + a(n - k + a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,\lfloor \frac{k-1}{2}\rfloor)$ edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for $k \geq 3$ odd and all $n \geq k$, every $n$-vertex $2$-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e(G) \leq \max\{h(n,k,3),h(n,k,\frac{k-3}{2})\}$. The upper bound for $e(G)$ here is tight.

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arXiv2017

A stability version for a theorem of Erdős on nonhamiltonian graphs

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$ and $e(n,d):= \max\{h(n,d),h(n, \left \lfloor \frac{n-1}{2} \right \rfloor)\}$. Because $h(n,d)$ is quadratic in $d$, there exists a $d_0(n)=(n/6)+O(1)$ such that $e(n,1)> e(n, 2)> \dots >e(n,d_0)=e(n, d_0+1)=\dots = e(n,\left \lfloor \frac{n-1}{2} \right \rfloor)$. A theorem by Erdős states that for $d\leq \left \lfloor \frac{n-1}{2} \right \rfloor$, any $n$-vertex nonhamiltonian graph $G$ with minimum degree $δ(G) \geq d$ has at most $e(n,d)$ edges, and for $d > d_0(n)$ the unique sharpness example is simply the graph $K_n-E(K_{\lceil (n+1)/2\rceil})$. Erdős also presented a sharpness example $H_{n,d}$ for each $1\leq d \leq d_0(n)$. We show that if $d< d_0(n)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $δ(G) \geq d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d) - e(n, d+1) = n - 3d - 2 \geq n/2$ whenever $d< d_0(n)-1$.

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arXiv2017

Extensions of a theorem of Erdős on nonhamiltonian graphs

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erdős proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree $δ(G) \geq d$ has at most $h(n,d)$ edges. He also provides a sharpness example $H_{n,d}$ for all such pairs $n,d$. Previously, we showed a stability version of this result: for $n$ large enough, every nonhamiltonian graph $G$ on $n$ vertices with $δ(G) \geq d$ and more than $h(n,d+1)$ edges is a subgraph of $H_{n,d}$. In this paper, we show that not only does the graph $H_{n,d}$ maximize the number of edges among nonhamiltonian graphs with $n$ vertices and minimum degree at least $d$, but in fact it maximizes the number of copies of any fixed graph $F$ when $n$ is sufficiently large in comparison with $d$ and $|F|$. We also show a stronger stability theorem, that is, we classify all nonhamiltonian $n$-graphs with $δ(G) \geq d$ and more than $h(n,d+2)$ edges. We show this by proving a more general theorem: we describe all such graphs with more than ${n-(d+2) \choose k} + (d+2){d+2 \choose k-1}$ copies of $K_k$ for any $k$.

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