A bipartite graph on \(n\) vertices, with \(n\) even, is called uniquely bi-pancyclic (UBPC) if it contains precisely one cycle of length \(2m\) for every \(2 \leq m \leq \frac{n}{2}\). In this note, using computer programs, we show that if \(32 \leq n \leq 56\), and \(n \neq 44\), then there are no UBPC graphs of order \(n\). We also present the six non-isomorphic UBPC graphs of order 44. This improves the recent results on UBPC graphs of order at most 30.

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.