This paper investigates vertex colorings of graphs such that some rainbow subgraph \(R\) and some monochromatic subgraph \(M\) are forbidden. Previous work focused on the case that \(R = M\). Here we consider the more general case, especially the case that \(M = K_2\).

Let \(R\) be a commutative ring with identity. For any integer \(k > 1\), an element is a \(k\)-zero divisor if there are distinct \(k\) elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let \(Z(R,k)\) denote the set of the \(k\)-zero divisors of \(R\). In this paper we consider rings which are not \(k\)-integral domains (i.e. \(Z(R,k)\) is nontrivial) with finite \(Z(R,k)\). We show that a uniform \(n\) exists such that \(a^n = 0\) for all elements \(a\) of the nil-radical \(N\) and deduce that a ring \(R\) which is not a \(k\)-integral domain with more than \(k\) minimal prime ideals and whose nil-radical is finitely generated is finite, if \(Z(R,k)\) is finite.

Recording actual user interactions with a system is often useful for testing software applications. User-session based test suites that contain records of such interactions often find a complementary set of faults compared to test suites created by testers. This work utilizes such test suites and presents a new prioritization method that extends the existing combinatorial two-way inter-window prioritization by introducing weights on the distance between windows. We examine how a window distance between a pair of the parameter-value tuples influences the fault detection effectiveness. We evaluate several approaches used to calculate weights. Results show improvement over the original two-way inter-window prioritization technique, while the comparison of different weighting approaches reveals that a negative linear weighting calculation generally performs better in our experiments. The study demonstrates that the distance between windows in a pair is an important factor to consider in test suite prioritization, and that distinguishing windows by their order in a test case also improves the fault detection rate compared to using window labels that were utilized in previous methods. This work provides motivation for future work to develop general n-way combinatorial distance-based prioritization methods that take into account space and processing time requirements to address potential issues with large test suites.

In this paper, we identify \(LW\) and \(OW\) graphs, find the minimum \(\lambda\) for decomposition of \(\lambda K_n\) into these graphs, and show that for all viable values of \(\lambda\), the necessary conditions are sufficient for \(LW\)- and \(OW\)-decompositions using cyclic decompositions from base graphs.

For a connected graph \(G = (V, E)\), its inverse degree is defined as
\(\sum_{v\in V} \frac{1}{d(v)}\). Using an upper bound for the inverse degree of a graph obtained by Cioaba in [6], in this note, we present new sufficient conditions for some Hamiltonian properties of graphs.

A cancellable number (CN) is a fraction in which a decimal digit can be removed (“cancelled”) in the numerator and denominator without changing the value of the number; examples include \(\frac{16}{64}\) where the \(6\) can be cancelled and \(\frac{49}{98}\) where the \(9\) can be cancelled. We provide explicit infinite families of CNs with certain properties, subject to the restriction that the numerator and denominator have equal decimal length and the cancellable digits “line up”, i.e., all cancellation lines are vertical. The properties in question are that the CNs contain one of the following: a cancellable nine, a cancellable zero, a sequence of adjacent cancellable zeros, sequences of adjacent cancellable zeros and nines of the same length, and sequences of adjacent cancellations for certain other digits.

A \(\mathrm{GDD}(n_1+n_2,3; \lambda_1,\lambda_2)\) is a group divisible design with two groups of sizes \(n_1\) and \(n_2\), where \(n_1 < n_2\), with block size \(3\) such that each pair of distinct elements from the same group occurs in \(\lambda_1\) blocks and each pair of elements from different groups occurs in \(\lambda_2\) blocks. We prove that the necessary conditions are sufficient for the existence of group divisible designs \(\mathrm{GDD}(n_1+n_2, 3; \lambda_1, \lambda_2)\) with equal number of blocks of configuration \((1, 2)\) and \((0, 3)\) for \(n_1 + n_2 \leq 20\), \(n_1 \neq 2\) and in general for \(n_1 = 1, 3, 4, n_2 – 1\), and \(n_2 – 2\).

Given nonnegative integers \(a\), \(b\), \(c\), and \(d\), the transition function \(\nabla\) is defined by \(\nabla(a, b, c, d) = (|a-b|, |b-c|, |c-d|, |d-a|)\). The Diffy problem asks if it can reach \((0, 0, 0, 0)\) after some iterations of \(\nabla\) on the four numbers. If \((a, b, c, d)\) can transfer to \((0, 0, 0, 0)\) iterated by \(\nabla\) operations, the smallest \(N\) such that \(\nabla^N(a, b, c, d) = (0, 0, 0, 0)\) is called the stopping steps of the Diffy problem. In this paper, we will show that there exists \(N\) such that \(\nabla^N(a, b, c, d) = (0, 0, 0, 0)\) and the loose upper bound and exact upper bound of \(N\). In addition, we will also show that we can find a starting vector \((a, b, c, d)\) so that it reaches the zero vector \((0, 0, 0, 0)\) after exact \(k\) steps for any given positive integer \(k\).