Gyarfas conjectured in 2011 that every $r$-edge-colored $K_n$ contains a monochromatic component of bounded (perhaps three) diameter on at least $n/(r-1)$ vertices. Letzter proved this conjecture with diameter four. In this note we improve the re
sult in the case of $r=3$: We show that in every $3$-edge-coloring of $K_n$ either there is a monochromatic component of diameter at most three on at least $n/2$ vertices or every color class is spanning and has diameter at most four.
In this paper, we show that, for all $ngeq 5$, the maximum number of $2$-chains in a butterfly-free family in the $n$-dimensional Boolean lattice is $leftlceilfrac{n}{2}rightrceilbinom{n}{lfloor n/2rfloor}$. In addition, for the height-2 poset $K_{
s,t}$, we show that, for fixed $s$ and $t$, a $K_{s,t}$-free family in the $n$-dimensional Boolean lattice has $Oleft(nbinom{n}{lfloor n/2rfloor}right)$ $2$-chains.
Given a hereditary property $mathcal H$ of graphs and some $pin[0,1]$, the edit distance function $operatorname{ed}_{mathcal H}(p)$ is (asymptotically) the maximum proportion of edits (edge-additions plus edge-deletions) necessary to transform any gr
aph of density $p$ into a member of $mathcal H$. For any fixed $pin[0,1]$, $operatorname{ed}_{mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $pin[0,1]$ for which infinitely many CRGs are required to compute $operatorname{ed}_{mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $operatorname{ed}_{mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.
For a fixed planar graph $H$, let $operatorname{mathbf{N}}_{mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle, the asymptotic value of $operatorname{mathbf{N}}_{mathcal{P}}(n,C
_m)$ is currently known for $min{3,4,5,6,8}$. In this note, we extend this list by establishing $operatorname{mathbf{N}}_{mathcal{P}}(n,C_{10})sim(n/5)^5$ and $operatorname{mathbf{N}}_{mathcal{P}}(n,C_{12})sim(n/6)^6$. We prove this by answering the following question for $min{5,6}$, which is interesting in its own right: which probability mass $mu$ on the edges of some clique maximizes the probability that $m$ independent samples from $mu$ form an $m$-cycle?
We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the
clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.
For a planar graph $H$, let $operatorname{mathbf{N}}_{mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $operatorname{mathbf{N}}_{mathcal P}(n,P_7)sim{4over 27}n^4$, $operatorname{
mathbf{N}}_{mathcal P}(n,C_6)sim(n/3)^3$, $operatorname{mathbf{N}}_{mathcal P}(n,C_8)sim(n/4)^4$ and $operatorname{mathbf{N}}_{mathcal P}(n,K_4{1})sim(n/6)^6$, where $K_4{1}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $operatorname{mathbf{N}}_{mathcal P}(n,P_{2m+1})$ and $operatorname{mathbf{N}}_{mathcal P}(n,C_{2m})$ for $mgeq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $operatorname{mathbf{N}}_{mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.
Given a hereditary property of graphs $mathcal{H}$ and a $pin [0,1]$, the edit distance function ${rm ed}_{mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficien
t to ensure that the resulting graph satisfies $mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $mathcal{H}$ and the $mathcal{H}$-chromatic number of a random graph. Let $mathcal{H}$ be the property of forbidding an ErdH{o}s-R{e}nyi random graph $Fsim mathbb{G}(n_0,p_0)$, and let $varphi$ represent the golden ratio. In this paper, we show that if $p_0in [1-1/varphi,1/varphi]$, then a.a.s. as $n_0toinfty$, begin{align*} {rm ed}_{mathcal{H}}(p) = (1+o(1)),frac{2log n_0}{n_0} cdotminleft{ frac{p}{-log(1-p_0)}, frac{1-p}{-log p_0} right}. end{align*} Moreover, this holds for $pin [1/3,2/3]$ for any $p_0in (0,1)$.
In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be split such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise disjoint union of $
n$ parts of size at most $k$ each such that there is an edge between any two distinct parts. Let $$ f(n,H) = min {k in mathbb N : mbox{there is an $(n,k)$-graph $G$ such that $H otsubseteq G$}} . $$ Barbanera and Ueckerdt observed that $f(n, H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Turan exponent, i.e., ${rm ex}(n, H) = Theta(n^r)$ for some $r$, we show that $Omega (n^{2/r -1}) = f(n, H) = O (n^{2/r-1} log ^{1/r} n)$. We extend this result to all bipartite graphs for which an upper and a lower Turan exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =Theta(n^{1/3})$ for any fixed $t$.
Let ${rm ex}_{mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${rm ex}_{mathcal{P}}(n,T,H)$ is the well studied function, the planar Turan number of
$H$, denoted by ${rm ex}_{mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${rm ex}_{mathcal{P}}(n,C_4)$ and ${rm ex}_{mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${rm ex}_{mathcal{P}}(n,C_6)leq frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${rm ex}_{mathcal{P}}(n,C_6) leq frac{5}{2}n-7$, for all $ngeq 18$, which improves Lans result. We also pose a conjecture on ${rm ex}_{mathcal{P}}(n,C_k)$, for $kgeq 7$.
Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path of length $k$, is $n^{{lfloor{frac{k}{2}}rfloor}+1}$. In this paper we determine the asymptotic value o
f $f(n,P_4)$ and give conjectures for longer paths.
