In this paper, we first present a combinatorial proof of the recurrence relation about the number of the inverse-conjugate compositions of \(2n+1\), \(n > 1\). And then we get some counting results about the inverse-conjugate compositions for special compositions. In particular, we show that the number of the inverse-conjugate compositions of \(4k+1\), \(k > 0\) with odd parts is \(2^k\), and provide an elegant combinatorial proof. Lastly, we give a relation between the number of the inverse-conjugate odd compositions of \(4k+1\) and the number of the self-inverse odd compositions of \(4k+1\).

In this study, by using Jacobsthal and Jacobsthal Lucas matrix sequences, we define \(k\)-Jacobsthal and \(k\)-Jacobsthal Lucas matrix sequences depending on one parameter \(k\). After that, by using two parameters \((s,t)\), we define \((s,t)\)-Jacobsthal and \((s,t)\)-Jacobsthal Lucas matrix sequences. And then, we establish combinatoric representations of all of these matrices.

A graph \(G\) is \(1\)-planar if it can be embedded in the plane \(\mathbb{R}^2\) so that each edge of \(G\) is crossed by at most one other edge. In this paper, we show that each \(1\)-planar graph of maximum degree \(\Delta\) at least \(7\) with neither intersecting triangles nor chordal \(5\)-cycles admits a proper edge coloring with \(\Delta\) colors.

Dirac showed that in a \((k-1)\)-connected graph there is a path through each \(k\) vertices. The path \(k\)-connectivity \(\pi_k(G)\) of a graph \(G\), which is a generalization of Dirac’s notion, was introduced by Hager in 1986. Recently, Mao introduced the concept of path \(k\)-edge-connectivity \(\omega_k(G)\) of a graph \(G\). Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\omega_4(G \circ H) \geq \omega_4(G) |V(H)|\) for any two graphs \(G\) and \(H\). Moreover, the bound is sharp.

A graph \(G = (V(G), E(G))\) is even graceful and equivalently graceful, if there exists an injection \(f\) from the set of vertices \(V(G)\) to \(\{0, 1, 2, 3, 4, \ldots, 2|E(G)|\}\) such that when each edge \(uv\) is assigned the label \(|f(u) – f(v)|\), the resulting edge labels are \(2, 4, 6, \ldots, 2|E(G)|\). In this work, we use even graceful labeling to give a new proof for necessary and sufficient conditions for the gracefulness of the cycle graph. We extend this technique to odd graceful and super Fibonacci graceful labelings of cycle graphs via some number theoretic concept, called a balanced set of natural numbers.

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every \(1\)-planar graph without \(4\)-cycles or adjacent \(5\)-vertices is \(5\)-colorable.

In previous researches on classification problems, there are some similar results obtained between \(f\)-coloring and \(g_c\)-coloring. In this article, the author shows that there always are coincident classification results for a regular simple graph \(G\) when the \(f\)-core and the \(g_c\)-core of \(G\) are same and \(f(v) = g(v)\) for each vertex \(v\) in the \(f\)-core (the \(g_c\)-core) of \(G\). However, it is not always coincident for nonregular simple graphs under the same conditions. In addition, the author obtains some new results on the classification problem of \(f\)-colorings for regular graphs. Based on the coincident correlation mentioned above, new results on the classification problem of \(g_c\)-colorings for regular graphs are deduced.

The integrity of a graph \(G = (V, E)\) is defined as \(I(G) = \min\{|S| + m(G-S): S \subseteq V(G)\}\), where \(m(G-S)\) denotes the order of the largest component in the graph \(G-S\). This is a better parameter to measure the stability of a network, as it takes into account both the amount of work done to damage the network and how badly the network is damaged. Computationally, it belongs to the class of intractable problems known as NP-hard. In this paper, we develop a heuristic algorithm to determine the integrity of a graph. Extensive computational experience on \(88\) randomly generated graphs ranging from \(20\%\) to \(90\%\) densities and from \(100\) to \(200\) vertices has shown that the proposed algorithm is very effective.