scholarly journals Note on Long Paths in Eulerian Digraphs

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Charlotte Knierim ◽  
Maxime Larcher ◽  
Anders Martinsson
Keyword(s):  
Recent Result ◽  
Short Note ◽  
Average Degree ◽  
Logarithmic Factor ◽  
Long Paths

Long paths and cycles in Eulerian digraphs have received a lot of attention recently. In this short note, we show how to use methods from [Knierim, Larcher, Martinsson, Noever, JCTB 148:125--148] to find paths of length $d/(\log d+1)$ in Eulerian digraphs with average degree $d$, improving  the recent result of $\Omega(d^{1/2+1/40})$. Our result is optimal up to at most a logarithmic factor.  

scholarly journals A short note on hit-and-miss hyperspaces

2003 ◽  
Vol 4 (2) ◽  
pp. 281
Author(s):  
René Bartsch ◽  
Harry Poppe
Keyword(s):  
Recent Result ◽  
Short Note

<p>Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “κ-Lindelöfness” and “κ-compactness” for cardinals κ. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of Som Naimpally, to who this article is hearty dedicated.</p>

scholarly journals Long Paths and Cycles Passing Through Specified Vertices Under the Average Degree Condition

2015 ◽  
Vol 32 (1) ◽  
pp. 279-295
Author(s):  
Binlong Li ◽  
Bo Ning ◽  
Shenggui Zhang
Keyword(s):  
Average Degree ◽  
Degree Condition ◽  
Long Paths

scholarly journals A Better Lower Bound on Average Degree of 4-List-Critical Graphs

10.37236/5971 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Short Note ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

This short note proves that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

scholarly journals Note on the Multicolour Size-Ramsey Number for Paths,

10.37236/7954 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Monochromatic Copy

The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = \Omega(r^2 n)$.

A NOTE ON A COMPLETE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2018 ◽  
Vol 98 (1) ◽  
pp. 60-63
Author(s):  
DIEGO MARQUES ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Recent Result ◽  
Real Number ◽  
Complete Solution ◽  
Short Note ◽  
Radius Of Convergence ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Functions

Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$. This solves the question posed by K. Mahler completely.

A note on a theorem of Baer

2018 ◽  
Vol 17 (03) ◽  
pp. 1850044
Author(s):  
Haoran Yu
Keyword(s):  
Recent Result ◽  
Short Note ◽  
Baer’S Theorem

In this short note, we generalize a recent result of Zhang [A generalization of Baer’s theorem, Comm. Algebra (2017), doi: 10.1080/00927872.2017.1287275].

scholarly journals Counting Independent Sets in Hypergraphs

2014 ◽  
Vol 23 (4) ◽  
pp. 539-550 ◽  
Author(s):  
JEFF COOPER ◽  
KUNAL DUTTA ◽  
DHRUV MUBAYI
Keyword(s):  
Recent Result ◽  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Average Degree ◽  
General Statement ◽  
Free Graph ◽  
Hereditary Properties ◽  
Image Position ◽  
Linear Hypergraph

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least ${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$ independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least ${e^{(c_1-o(1)) \sqrt{n} \ln n}}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$. Further, we show that for all n, there exists a triangle-free graph with n vertices which has at most ${e^{(c_2+o(1))\sqrt{n}\ln n}}$ independent sets, where $c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$. This disproves a conjecture from [8].Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least ${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$ This is tight apart from the constant ck and generalizes a result of Duke, Lefmann and Rödl [9], which guarantees the existence of an independent set of size $\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$ Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs.

scholarly journals Average Degree Conditions Forcing a Minor

10.37236/5321 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Daniel J. Harvey ◽  
David R. Wood
Keyword(s):  
Recent Result ◽  
Complete Graph ◽  
Open Problem ◽  
Complete Bipartite Graph ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Arbitrary Graph ◽  
A Minor ◽  
High Degree

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph.

scholarly journals The Extremal Function for Cycles of Length $\ell$ mod $k$

10.37236/6257 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Benny Sudakov ◽  
Jacques Verstraete
Keyword(s):  
Bipartite Graph ◽  
Extremal Function ◽  
Absolute Constant ◽  
Complete Bipartite Graph ◽  
Short Note ◽  
Upper Bounds ◽  
Average Degree ◽  
Free Graph ◽  
The Extremal Function

Burr and Erdős conjectured that for each $k,\ell \in \mathbb Z^+$ such that $k \mathbb Z + \ell$ contains even integers, there exists $c_k(\ell)$ such that any graph of average degree at least $c_k(\ell)$ contains a cycle of length $\ell$ mod $k$. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on $c_k(\ell)$ appear in the literature. In this short note, for $1 \leq \ell \leq k$, we show that $c_k(\ell)$ is proportional to the largest average degree of a $C_{\ell}$-free graph on $k$ vertices, which determines $c_k(\ell)$ up to an absolute constant. In particular, using known results on Turán numbers for even cycles, we obtain $c_k(\ell) = O(\ell k^{2/\ell})$ for all even $\ell$, which is tight for $\ell \in \{4,6,10\}$. Since the complete bipartite graph $K_{\ell - 1,n - \ell + 1}$ has no cycle of length $2\ell$ mod $k$, it also shows $c_k(\ell) = \Theta(\ell)$ for $\ell = \Omega(\log k)$.

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