Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a, b ∈ I and x ∈ V \ I , {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V , and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable...

It has been showed in [4] that any bipartite graph Ka,b, where a, b are even is decomposable into closed trails of prescribed even lengths. In this article we consider the corresponding question for open trails. We prove a necessary and sufficient condition for graphs Ka,b to be decomposable into edge-disjoint open trails of positive lengths (less than ab) whenever these lengths sum up to the s...

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is nicely locally clique-width-decomposable. This notion generalizes that of a nicely locally tree-decomposable class. The graphs of s...

Abstract Assume that k is an algebraically closed field and A a finite-dimensional wild -algebra. Recently, L. Gregory M. Prest proved in this case the width of lattice all pointed -modules undefined. Hence result Ziegler implies there exists super-decomposable pure-injective -module, if base countable. Here we give different straightforward proof fact. Namely, show special family -modules, cal...

A graph is magic if the edges can be labeled with nonnegative real numbers such that (i) different edges have distinct labels, and (ii) the sum of the labels of edges incident to each vertex is the same. Regular magic graphs are characterized herein by the nonappearance of certain bipartite subgraphs. This implies that line connectivity is a crucial property for characterizing regular magic gra...

Let G be a graph of order n and r , 1 ≤ r ≤ n, a fixed integer. G is said to be r -vertex decomposable if for each sequence (n1, . . . , nr ) of positive integers such that n1 + · · · + nr = n there exists a partition (V1, . . . , Vr ) of the vertex set of G such that for each i ∈ {1, . . . , r}, Vi induces a connected subgraph of G on ni vertices. G is called arbitrarily vertex decomposable if...

There are many results on edge-magic, and vertex-magic, labellings of finite graphs. Here we consider magic labellings of countably infinite graphs over abelian groups. We also give an example of a finite connected graph that is edge-magic over one, but not over all, abelian groups of the appropriate order.

We consider the dihamiltonian decomposition problem for 3regular graphs. A graphG is dihamiltonian decomposable if in the digraph obtained fromG by replacing each edge of G as two directed edges, the set of edges are partitioned into 3 edge-disjoint directed hamiltonian cycles. We suggest some conditions for dihamiltonian decomposition of 3-regular graphs: for a 3-regular graph G, it is dihamil...

A graph G = (V, E) is said to be magic if there exists an integer labeling f : V ∪ E −→ [1, |V ∪ E|] such that f(x) + f(y) + f(xy) is constant for all edges xy ∈ E. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n + o(n) which contain a complete graph of order n. Bounds on Sidon sets show that the order of such a graph is at least n + o(n). We close the gap ...

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑ u∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it c...