A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number ${\chi }_f\left(G\right)$ of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, $\chi \left(G\right)\le {\chi }_f\left(G\right)$. In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and ${\chi }_f\left(G\right)$. We also obtain the smallest non-fall colorable...

A consecutive colouring of a graph is a proper edge colouring with posi- tive integers in which the colours of edges incident with each vertex form an interval of integers. The idea of this colouring was introduced in 1987 by Asratian and Kamalian under the name of interval colouring. Sevast- janov showed that the corresponding decision problem is NP-complete even restricted to the class of bipartite graphs. We focus our attention on the class of consecutively colourable graphs whose all induced...

The Erdős-Faber-Lovász conjecture is the statement that every graph that is the union of n cliques of size n intersecting pairwise in at most one vertex has chromatic number n. Kahn and Seymour proved a fractional version of this conjecture, where the chromatic number is replaced by the fractional chromatic number. In this note we investigate similar fractional relaxations of the Erdős-Faber-Lovász conjecture, involving variations of the fractional chromatic number. We exhibit some relaxations that...

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P,Q)-total...

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let [...] be an additive hereditary property of graphs. A [...] -edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property [...] . In this paper we present some results on fractional [...] -edge-colorings. We determine the fractional [...] -edge chromatic number for matroidal properties of graphs.

This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revolutionizes communication. Traditional vertex coloring provides a conceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various extensions to cover technical and organizational side constraints. Among these ramifications are T-coloring and...

Starting from the famous Königsberg bridge problem which Euler described in 1736, we intend to show that some results obtained 180 years later by König are very close to Euler's discoveries.

Let $G_1$ and $G_2$ be copies of a graph $G$, and let $f:V\left(G_1\right)\to V\left(G_2\right)$ be a function. Then a functigraph $C(G,f)=(V,E)$ is a generalization of a permutation graph, where $V=V\left(G_1\right)\cup V\left(G_2\right)$ and $E=E\left(G_1\right)\cup E\left(G_2\right)\cup \{uv:u\in V\left(G_1\right),v\in V\left(G_2\right),v=f\left(u\right)\}$. In this paper, we study colorability and planarity of functigraphs.