A Hamiltonian graph \(G\) is said to be \(\ell\)-path-Hamiltonian, where \(\ell\) is a positive integer less than or equal to the order of \(G\), if every path of order \(\ell\) in \(G\) is a subpath of some Hamiltonian cycle in \(G\). The Hamiltonian cycle extension number of \(G\) is the maximum positive integer \(L\) for which \(G\) is \(\ell\)-path-Hamiltonian for every integer \(\ell\) with \(1 \leq \ell \leq L\). Hamiltonian cycle extension numbers are determined for several well-known cubic Hamiltonian graphs. It is shown that if \(G\) is a cubic Hamiltonian graph with girth \(g\), where \(3 \leq g \leq 7\), then \(G\) is \(\ell\)-path-Hamiltonian only if \(1 \leq \ell \leq g\).

In a red-blue coloring of a graph \(G\), every edge of \(G\) is colored red or blue. For two graphs \(F\) and \(H\), the Ramsey number \(R(F, H)\) of \(F\) and \(H\) is the smallest positive integer \(n\) such that every red-blue coloring of the complete graph \(K_n\) of order \(n\) results in either a subgraph isomorphic to \(F\) all of whose edges are colored red or a subgraph isomorphic to \(H\) all of whose edges are colored blue. While the study of Ramsey numbers has been a popular area of research in graph theory, over the years a number of variations of Ramsey numbers have been introduced. We look at several of these, with special emphasis on some of those introduced more recently.

For a graph \(G\) of size \(m\), a graceful labeling of \(G\) is an injective function \(f : V(G) \to \{0, 1, \dots, m\}\) that gives rise to a bijective function \(f’ : E(G) \to \{1, 2, \dots, m\}\) defined by \(f'(uv) = |f(u) – f(v)|\). A graph \(G\) is graceful if \(G\) has a graceful labeling. Over the years, a number of variations of graceful labelings have been introduced, some of which have been described in terms of colorings. We look at several of these, with special emphasis on some of those introduced more recently.

For a connected graph \(G\) of order at least \(3\), let \(c : E(G) \to \{1, 2, \dots, k\}\) be an edge coloring of \(G\) where adjacent edges may be colored the same. Then \(c\) induces a vertex coloring \(c’\) of \(G\) by assigning to each vertex \(v\) of \(G\) the set of colors of the edges incident with \(v\). The edge coloring \(c\) is called a majestic \(k\)-edge coloring of \(G\) if the induced vertex coloring \(c’\) is a proper vertex coloring of \(G\). The minimum positive integer \(k\) for which a graph \(G\) has a majestic \(k\)-edge coloring is the majestic chromatic index of \(G\) and denoted by \(\chi_{m}^{‘} (G)\). For a graph \(G\) with \(\chi_{m}^{‘}(G) = k\), the minimum number of distinct vertex colors induced by a majestic \(k\)-edge coloring is called the majestic chromatic number of \(G\) and denoted by \(\psi(G)\). Thus, \(\psi(G)\) is at least as large as the chromatic number \(\chi(G)\) of a graph \(G\). Majestic chromatic indexes and numbers are determined for several well-known classes of graphs. Furthermore, relationships among the three chromatic parameters \(\chi_m(G)\), \(\psi(G)\), and \(\chi(G)\) of a graph \(G\) are investigated.

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.