A graph \(G\) with vertex set \(V\) and edge set \(E\) is called super edge-graceful if there is a bijection \(f\) from \(E\) to \(\{0, \pm 1, \pm 2, \dots, \pm (|E| – 1)/2\}\) when \(|E|\) is odd and from \(E\) to \(\{\pm1, \pm2, \dots, \pm|E|/2\}\) when \(|E|\) is even, such that the induced vertex labeling \(f^*\) defined by \(f^*(u) = \sum f(uv)\) over all edges \(uv\) is a bijection from \(V\) to \(\{0, \pm 1, \pm 2, \dots, \pm (|V| – 1)/2\}\) when \(|V|\) is odd and from \(V\) to \(\{\pm1, \pm2, \dots, \pm|V|/2\}\) when \(|V|\) is even. A kite is a graph formed by merging a cycle and a path at an endpoint of the path. In this paper, we prove that all kites with \(n \geq 5\) vertices, \(n \neq 6\), are super edge-graceful.

A graph with \(v\) vertices is \((r)\)-pancyclic if it contains precisely \(r\) cycles of every length from \(3\) to \(v\). A bipartite graph with an even number of vertices \(v\) is said to be \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from \(4\) to \(v\). A bipartite graph with an odd number of vertices \(v\) and minimum degree at least \(2\) is said to be oddly \((r)\)-bipancyclic if it contains precisely \(r\) cycles of each even length from \(4\) to \(v-1\). In this paper, using a computer search, we classify all \((r)\)-pancyclic and \((r)\)-bipancyclic graphs, \(r \geq 2\), with \(v\) vertices and at most \(v+5\) edges. We also classify all oddly \((r)\)-bipancyclic graphs, \(r \geq 1\), with \(v\) vertices and at most \(v+4\) edges.

A handicap distance antimagic labeling of a graph \(G = (V,E)\) with \(n\) vertices is a bijection \(f : V \to \{1,2,\dots,n\}\) with the property that \(f(x_i) = i\) and the sequence of the weights \(w(x_1), w(x_2), \dots, w(x_n)\) (where \(w(x_i) = \sum_{x_j \in N(x_i)}{f(x_j)}\)) forms an increasing arithmetic progression. A graph \(G\) is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling.

We construct regular handicap distance antimagic graphs for every feasible odd order.

A well-known subclass of chordal graphs is formed by proper interval graphs. Due to their very special structural properties, several problems proved hard to solve for interval graphs can have better solutions for this subclass. In this paper, we address the recognition problem, proposing an update of one of the first existing linear algorithms. The outcome is a simple and efficient algorithm. In addition, we present a certifying algorithm for the recognition of proper interval graphs

A bipartite graph on \(2n\) vertices is called bipancyclic if it contains cycles of every length from \(4\) to \(2n\). In this paper we address the question: what is the minimum number of edges in a bipancyclic graph? We present a simple analysis of some small orders using chord patterns.

A vertex set \(U \subset V\) in a connected graph \(G = (V, E)\) is a cutset if \(G – U\) is disconnected. If no proper subset of \(U\) is also a cutset of \(G\), then \(U\) is a minimal cutset. An \(\mathcal{MVC}\)-partition \(\pi = \{V_1, V_2, \dots, V_k\}\) of the vertex set \(V(G)\) of a connected graph \(G\) is a partition of \(V(G)\) such that every \(V_i \in \pi\) is a minimal cutset of \(G\). For an \(\mathcal{MVC}\)-partition \(\pi\) of \(G\), the \(\pi\)-graph \(G_{\pi}\), of \(G\) has vertex set \(\pi\) such that \(V’,V” \in \pi\) are adjacent in \(G_{\pi}\), if and only if there exist \(v’ \in V’\) and \(v” \in V”\) such that \(v’v” \in E(G)\). Graphs that are a \(\pi\)- graphs of cycles are characterized. A homomorphic image \(H\) of a graph \(G\) can be obtained from a partition \(\mathcal{P} = {V_1, V_2,…, V_k}\) of \(V(G)\) into independent sets such that \(V(H) = {v_1,v_2,…, v_k}\), where \(v_i\) is adjacent to \(v_j\); if and only if some vertex of \(V_j\) is adjacent to some vertex of \(V_j\) in \(G\). By investigating graphs \(H\) that are homomorphic images of the Cartesian product \(H\Box K_2\), it is shown that for every nontrivial connected graph \(H\) and every integer \(r \geq 2\), there exists an \(r\)-regular graph \(G\) such that \(H\) is a homomorphic image of \(G\). It is also shown that every nontrivial tree \(T\) is a homomorphic image of \(T\Box K_2\) but that not all graphs \(H\) are homomorphic images of \(H\Box K_2\).

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.

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 \(k\).