Abs t r ac t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordina...

The Ramsey number r(F, G) is the least number N such that in every two-coloring (R, B)=(red, blue) of the edges of KN , either there is a red copy of F or else a blue copy of G . The inequality (1) holds in view of the fact that the edges of the complete graph of order (m-1) (n-1) +s-1 can be given the coloring in which the blue graph, denoted (B), is isomorphic to (m-1)K„_~UKS_~ . Then the red...

We prove a conjecture of F uredi and Mubayi: For any graph G on n vertices with minimum degree r, there exists a two-coloring of the vertices of G with colors +1 and ?1, such that the closed neighborhood of each vertex contains more +1's than ?1's, and altogether the number of 1's does not exceed the number of ?1's by more than O(n= p r). As a construction by F uredi and Mubayi shows, this is a...

Let Γ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of Γ turn the graph into deterministic complete automaton. Let the word s be a word in the alphabet of colors (considered also as letters) on the edges of Γ and let Γs be a mapping of vertices Γ. A coloring is called k-synchronizing if for any word t |Γt| ≥ k ...

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be two additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r s -fractional (P,Q)-total coloring of a finite graphG = (V,E) is a mapping f , which assigns an s-element subset of the set {1, 2, . . . , r} to each vert...

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Motivated results related to Cartesian products graphs, we initiate the study DP-chromatic number, $\chi_{DP}$, same. We show that $\chi_{DP}(G \square H) \leq \text{min}\{\chi_{DP}(G) + \text{col}(H), \chi_{DP}(H) \text{col}(G) \} - 1$ where $\text{col}(H)$ nu...

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Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [Zr] be the set of all s-element subsets of Zr. An (r, s)-fractional This work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by the Slovak VEGA Grant 1/0652/12. 2 A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R. Soták (P,Q)-total coloring of G is an a...