let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

A set S ⊆ V is called an q-set (q−-set, respectively) if S has at least two vertices and, for every u ∈ S, there exists v ∈ S, v 6= u such that N(u) ∩ N(v) 6= ∅ (N−(u)∩N−(v) 6= ∅, respectively). A digraph D is called s-quadrangular if, for every q-set S, we have | ∪ {N+(u) ∩ N(v) : u 6= v, u, v ∈ S}| ≥ |S| and, for every q−set S, we have | ∪ {N−(u) ∩ N−(v) : u, v ∈ S)} ≥ |S|. We conjecture that...

we show that in {bf zf} set theory without choice, the ultrafilter mbox{principle} ({bf up}) is equivalent to several compactness theorems for alexandroff discrete spaces and to rudin's lemma, a basic tool in topology and the theory of quasi-continuous domains. important consequences of rudin's lemma are various lift lemmas, saying that certain properties of posets are inherited by th...

In this paper we calculate the cardinality of the set Sn(T, τ) of all permutations in Sn that avoid one pattern from S4 and a nonempty set of patterns from S3. The main body of the paper is divided into four sections corresponding to the cases |T | = 1, 2, 3, and |T | ≥ 4. At the end of each section we provide the tables accumulating all the results obtained.

Given a PL (piecewise linear) set S in the plane R we consider the set Pε(S) consisting of all pixels of size ε that touch S. This pixelation Pε(S) resembles the original set, but may not well approximate various important invariants of the original set such as Betti numbers, perimeter, curvature measures. We describe an algorithm that associates to the pixelation Pε(S) a PL-set Pε(S) which app...

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A set system is a pair S = (V (S),∆(S)), where ∆(S) is a family of subsets of the set V (S). We refer to the members of ∆(S) as the stable sets of S. A homomorphism between two set systems S and T is a map f : V (S) → V (T ) such that the preimage under f of every stable set of T is a stable set of S. Inspired by a recent generalization due to Engström of Lovász’s Hom complex construction, the ...

We study the problem k-SET SPLITTING in fixed parameter complexity. We show that the problem can be solved in time O∗(2.6494k), improving on the best currently known running time of O∗(8k). This is done by showing that a non-trivial instance must have a small minimal SET COVER, and using this to reduce the problem to a series of small instances of MAX SAT. We also give a linear kernel containin...