As the counterpart of classical theorems on cycles consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed following open problem 2008: What is maximum C $C$ such that for all positive ε < $\varepsilon \lt C$ sufficiently large n $n$ , every G $G$ order with radius ρ ( ) > ⌊ 2 4 ⌋ $\rho (G)\gt \sqrt{\lfloor \frac{{n}^{2}}{4}\rfloor }$ contains a cycle length ℓ...

The channel assignment problem involves assigning radio channels to transmitters, using a small span of channels but without causing excessive interference. We consider a standard model for channel assignment, the constraint matrix model, which extends ideas of graph colouring. Given a graph G = (V; E) and a length l(uv) for each edge uv of G, we call an assignment : V → {1; : : : ; t} feasible...

Given a graph whose edges are coloured, on how many vertices can we find a monochromatic subgraph of a certain type, such as a connected subgraph, or a cycle, or some type of tree? Also, how many such monochromatic subgraphs do we need so that their vertex sets form either a partition or a covering of the vertices of the original graph? What happens for the analogous situations for hypergraphs?...

In 1893 J. J. Sylvester [8] posed the following celebrated problem: Given a finite collection of points in the affine plane, not all lying on a line, show that there exists a line which passes through precisely two of the points. Sylvester’s problem was reposed by Erdős in 1944 [4] and later that year a proof was given by Gallai [6]. Since then, many proofs of the Sylvester-Gallai Theorem have ...

In 1963, Tibor Gallai [9] asked whether every strongly connected directed graph D is spanned by α directed circuits, where α is the stability of D. We give a proof of this conjecture. 1 Coherent cyclic orders. In this paper, circuits of length two are allowed. Since loops and multiple arcs play no role in this topic, we will simply assume that our digraphs are loopless and simple. A directed gr...