A division of a cake \(X = X_1 \cup \cdots \cup X_n\) among \(n\) players with associated probability measures \(\mu_1, \ldots, \mu_n\) on \(X\) is said to be:

(a) exact in the ratios of \(\alpha_1 : \alpha_2 : \cdots : \alpha_n\) provided whenever \(1 \leq i, j \leq n\), \(\frac{\mu_i(X_j)} { \mu_1(X)} = \alpha_i / (\alpha_1 + \cdots + \alpha_n)\)

(b) \(\epsilon\)-near exact in the ratios \(\alpha_1 : \alpha_2 : \cdots : \alpha_n\) provided whenever \(1 \leq i, j \leq n\), \(|\frac{\mu_i(X_i)}{\mu_1(X_1)} + \cdots +\frac {\alpha_j}{\alpha_1 + \cdots + \alpha_n}| < \epsilon\)

(c) envy free in ratios \(\alpha_1 : \alpha_2 : \cdots : \alpha_n\) provided whenever \(1 \leq i, j \leq n\), \(\frac{\mu_i(X_i)}{\mu_i(X_j)} \geq \frac{\alpha_i}{\alpha_j}\).

A moving knife exact division is described for two players and it is shown there can be no finite exact algorithm for \(n \geq 2\) players. A bounded finite \(\epsilon\)-near exact algorithm is given which is used to produce a finite envy free, \(\epsilon\)-near exact algorithm.

We study bounds on the cardinality of sum-distinct sets of \(n\)-vectors with nonnegative integral components under component-wise real-number addition. A subclass of sum-distinct sets induced by an \(n \times n\) integral matrix of rank \(n\) is studied as well.

A family of subsets satisfies the Helly property when every subfamily of it, formed by pairwise intersecting subsets, has a common element. A graph is clique-Helly when the family of subsets of vertices which induces the maximal cliques of the graph satisfies the Helly property. We describe a characterization of clique-Helly graphs, leading to a polynomial time algorithm for recognizing them.

A semi-complete bigraph \(G\) has adjacency matrix
\[A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix},\]
where \(B + B^T = J – I\), so \(B\) is the adjacency matrix of a tournament \(T\) corresponding to \(G\). We determine algebraic and structural properties of this class of graphs. In particular, we obtain a condition equivalent to the connectedness of a semi-complete bigraph; moreover we determine characterizations of semi-complete bigraphs having 4 distinct eigenvalues in the case of \(G\) regular or \(A\) irreducible. In particular, a regular semi-complete bigraph has 4 distinct eigenvalues if and only if it corresponds to a doubly regular tournament.

Let \(D\) be an asymmetric digraph and \(A\) a finite group. We give a formula for the characteristic polynomial of a cyclic \(A\)-cover of \(D\). This is a generalization of a formula for the characteristic polynomial of a regular covering of a graph. Furthermore, we discuss cyclic abelian covers of \(D\).