Gyárfas proved that every coloring of the edges $K_n$ with $t+1$ colors contains a monochromatic connected component size at least $n/t$. Later, Gyárfás and Sárközy asked for which values $\gamma=\gamma(t)$ does following strengthening almost complete graphs hold: if $G$ is an $n$-vertex graph minimum degree $(1-\gamma)n$, then $(t+1)$-edge We show $\gamma= 1/(6t^3)$ suffices, improving result ...

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms can preserve it. The number $G$, denoted by $D(G)$, the minimum colors required for such coloring, and threshold $\theta(G)$, $k$ that every $k$-coloring distinguishing. As an alternative definition, $\theta(G)$ one more than maximum cycles in cycle decomposition $G$. In this paper, we characterize $\thet...

We want to provide some background from coding theory and geometry. Let C be a binary linear code of length N, dimension k, and minimum distance at least 4. Let G be a generator matrix for C of size k_N. Then C has length N and dimension N&k. Put N&k=n+1. A basis for C gives a matrix M of size (n+1)_N. Since C has minimum distance at least 4 it follows that the columns of M form a set S of N po...

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is minimum integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red or blue $H$. Let $G+H$ be graph obtained from vertex disjoint $H$ by adding new edges connecting completely, $F_m=K_1+mK_2$ $B_p(n)=K_p+nK_1$. It shown $R(F_m,B_p(n))=2(n+p-1)+1$ for fixed $m, p$ large $n$.

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow cycle (that is, set edges from distinct $O_i$'s, forming an cycle). As part proof, we characterize those families $K_{n+1}$ do not have any cycle. also $K_{n+1}$, as well edge-disjoint nonempty subgraphs without

in this paper, we introduce the concept of semihyperring $(r,+,cdot)$ together with a suitable partial order $le$. moreover, we introduce and study hyperideals in ordered semihyperrings. simple ordered semihyperrings are defined and its characterizations are obtained. finally, we study some properties of quasi-simple and $b$-simple ordered semihyperrings.