A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each...

We show that if n is a positive integer and $2^{\aleph ₀}\le \aleph ₙ$, then for every positive integer m and for every real constant c > 0 there are functions $f₁,...,f_{n+m}:ℝⁿ\to ℝ$ such that $(f₁,...,f_{n+m})\left(ℝⁿ\right)={ℝ}^{n+m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i₁},...,f_{iₙ})\left(y\right)=y+w$ for $y\in x+(-c,c)\times {ℝ}^{n-1}$.

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple graph and $E_G\left(v\right)$ denote the set of edges incident with a vertex $v$. A neighbor sum distinguishing (NSD) total coloring $\phi$ of $G$ is a proper total coloring of $G$ such that ${\sum }_{z\in E_G\left(u\right)\cup \left\{u\right\}}\phi \left(z\right)\ne {\sum }_{z\in E_G\left(v\right)\cup \left\{v\right\}}\phi \left(z\right)$ for each edge $uv\in E\left(G\right)$. Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $\Delta$ admits an NSD total $(\Delta +3)$-coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $\Delta \ge 11$ but without $5$-cycles by applying the Combinatorial Nullstellensatz.

This note presents the study of the conjugacy classes of elements of some given finite order $n$ in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if $n$ is even, $n=3$ or $n=5$, and that it is equal to $3$ (respectively $9$) if $n=9$ (respectively if $n=15$) and to $1$ for all remaining odd orders. Some precise representative elements of the classes are given.

The generalized $k$-connectivity ${\kappa }_k\left(G\right)$ of a graph $G$ was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as ${\lambda }_k\left(G\right)=min\{\lambda \left(S\right):S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$, where $\lambda \left(S\right)$ denotes the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,...,T_{\ell }$ in $G$ such that $S\subseteq V\left(T_i\right)$ for $1\le i\le \ell$. In this paper we prove that for any two connected graphs $G$ and $H$ we have ${\lambda }_3(G\square H)\ge {\lambda }_3\left(G\right)+{\lambda }_3\left(H\right)$, where $G\square H$ is the Cartesian product of $G$ and $H$. Moreover, the bound is sharp. We also...

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................