New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number $R_D(s,t,d)$. The distance Ramsey number $R_D(s,t,d)$ is the minimum number $n$ such that for any graph $G$ on $n$ vertices, either $G$ contains an induced $s$-vertex subgraph isomorphic to a distance graph in ${\mathbb{R}}^d$ or $\stackrel{?}{G}$ contains an induced $t$-vertex subgraph isomorphic to the distance graph in ${\mathbb{R}}^d$. We obtain the upper and lower bounds on $R_D(s,s,d)$, which are similar to the bounds for the classical Ramsey number $R(?\frac{s}{[d/2]}?,?\frac{s}{[d/2]}?)$.     

Domination and upper domination of direct product graphs

Let $X_{\mathbb{Z}∕n\mathbb{Z}}$ denote the unitary Cayley graph of $\mathbb{Z}∕n\mathbb{Z}$. We present results on the tightness of the known inequality $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)$, where $\gamma$ and${\gamma }_t$ denote the domination number and total domination number, respectively, and $g$ is the arithmetic function known as Jacobsthal’s function. In particular, we construct integers $n$ with arbitrarily many distinct prime factors such that $\gamma \left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le {\gamma }_t\left(X_{\mathbb{Z}∕n\mathbb{Z}}\right)\le g\left(n\right)?1$. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.     

On the trivectors of a 6-dimensional symplectic vector space. II

Let V be a 6-dimensional vector space over a field $\mathbb{F}$, let f be a nondegenerate alternating bilinear form on V and let $\mathit{Sp}(V\text{,}f)?{\mathit{Sp}}_6(\mathbb{F})$ denote the symplectic group associated with $(V\text{,}f)$. The group $\mathit{GL}(V)$ has a natural action on the third exterior power ${?}^{3}V$ of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field $\mathbb{F}$. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of $\mathbb{F}$ that are contained in a fixed algebraic closure $\overline{\mathbb{F}}$ of $\mathbb{F}$. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of $\mathit{Sp}(V\text{,}f)?\mathit{GL}(V)$ on ${?}^{3}V$.     

On prisms,Möbius ladders and the cycle space of dense graphs

For a graph $G$, let $\left|G\right|$ denote its number of vertices, $\delta \left(G\right)$ its minimum degree and $Z_1(G;{\mathbb{F}}_2)$ its cycle space. Call a graph Hamilton-generated if and only if every cycle in $G$ is a symmetric difference of some Hamilton circuits of $G$. The main purpose of this paper is to prove: for every $\gamma >0$ there exists $n_0\in \mathbb{Z}$ such that for every graph $G$ with $\left|G\right|⩾n_0$ vertices,

• (1)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is odd, then $G$ is Hamilton-generated,
• (2)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is even, then the set of all Hamilton circuits of $G$ generates a codimension-one subspace of $Z_1(G;{\mathbb{F}}_2)$ and the set of all circuits of $G$ having length either $\left|G\right|-1$ or $\left|G\right|$ generates all of $Z_1(G;{\mathbb{F}}_2)$,
• (3)if $\delta \left(G\right)⩾(\frac{1}{4}+\gamma )\left|G\right|$ and $G$ is balanced bipartite, then $G$ is Hamilton-generated.

All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [I.B.-A. Hartman, Long cycles generate the cycle space of a graph, European J. Combin. 4 (3) (1983) 237–246] originates with A. Bondy.     

Gap vertex-distinguishing edge colorings of graphs

In this paper, we study a new coloring parameter of graphs called the gap vertex-distinguishing edge coloring. It consists in an edge-coloring of a graph $G$ which induces a vertex distinguishing labeling of $G$ such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distinguishing edge coloring of $G$ is called the gap chromatic number of $G$ and is denoted by $\mathrm{gap}\left(G\right)$.We here study the gap chromatic number for a large set of graphs $G$ of order $n$ and prove that $\mathrm{gap}\left(G\right)\in \{n?1,n,n+1\}$.     

Global H1 solvability of the 3D logarithmic Schrödinger equation

In this paper we investigate the existence of a unique global mild solution in $H^1\left({\mathbb{R}}^3\right)$ of the initial-boundary value problem associated with the logarithmic Schrödinger equation $i{?}_t\psi =?D\Delta \psi +\sigma \log \left(\right|\psi {|}^2)\psi$, with $D>0$ and $\sigma \in \mathbb{R}?\left\{0\right\}$.     

G-Ham Sandwich Theorems: Balancing measures by finite subgroups of spheres

Equivariant Ham Sandwich Theorems are obtained for the classical algebras $\mathbb{F}=\mathbb{R},\mathbb{C}\text{, and}\mathbb{H}$ and the finite subgroups G of their unit spheres. Given any n $\mathbb{F}$-valued Borel measures on ${\mathbb{F}}^n$ and any n-dimensional free $\mathbb{F}$-unitary representation of G, it is shown that there exists a Voronoi partition of ${\mathbb{F}}^n$ naturally determined by G which “G-balances” each measure, as realized by the simultaneous vanishing of each “G-average” of the measures of the partition?s isometric fundamental domains. Applications for real measures follow, among them that any n signed mass distributions on ${\mathbb{C}}^{(p?1)n/2}$ can be equipartitioned by a single complex regular p-fan if p is an odd prime.