An even 2-factor is one such that each cycle of length. A 4- regular graph G 4-edge-colorable if and only has two edge-disjoint 2- factors whose union contains all edges in G. It known the line a cubic without 3-edge-coloring not 4-edge-colorable. Hence, we are interested whether those graphs have an 2-factor. Bonisoli Bonvicini proved connected with number 2-factor, perfect matching [Even cycl...

The starting point of our work is a Ramsey-type theorem of Galvin (unpublished) which asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes (or even classes which have the property of Baire) then there is a perfect set P such that all pairs from P lie in the same class. The obvious generalization to n-tuples for n ≥ 3 is false. For example, look at the co...

We investigate the coloring properties of mixed interval hypergraphs having two families of subsets: the edges and the co-edges. In every edge at least two vertices have different colors. The notion of a co-edge was introduced recently in [2,3]: in every such a subset at least two vertices have the same color. The upper (lower) chromatic number is defined as a maximum (minimum) number of colors...

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

{sl let $[n]={1,dots, n}$ be colored in $k$ colors. a rainbow ap$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. conlon, jungi'{c} and radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow ap(4) free, when $n$ is even. based on their construction, we show that such a coloring of ...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

an acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. the acyclic chromatic index of a graph $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...

We prove that, for any pair of integers k, l ≥ 1, there exists an integerN(k, l) such that every graph with chromatic number at least N(k, l) contains either Kk or an induced odd cycle of length at least 5 or an induced cycle of length at least l. Given a graph with large chromatic number, it is natural to ask whether it must contain induced subgraphs with particular properties. One possibility...

We present a fairly general method for nding deterministic constructions obeying what we call krestrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 k con gurations appear) and families of...

Gallai-colorings of complete graphs – edge colorings such that no triangle is colored with three distinct colors – occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result – any Gallai-colored complete gr...