This paper is based on the splitting operation for binary matroids that was introduced by Raghunathan, Shikare and Waphare [ Discrete Math. \(184 (1998)\), p.\(267-271\)] as a natural generalization of the corresponding operation in graphs. Here, we consider the problem of determining precisely which graphs \(G\) have the property that the splitting operation, by every pair of edges, on the cycle matroid \(M(G)\) yields a graphic matroid. This problem is solved by proving that there are exactly four minor-minimal graphs that do not have this property.

In this paper, we study the connection of number theory with graph theory via investigating some uncharted properties of the directed graph \(\Gamma'(n)\) whose vertex set is \(\mathbb{Z}_n = \{0,1,\ldots,n-1\}\), and for which there is a directed edge from \(a \in \mathbb{Z}_n\) to \(b \in \mathbb{Z}_n\) if and only if \(a^3 \equiv b \pmod{n}\). For an arbitrary prime \(p\), the formula for the decomposition of the graph \(\Gamma(p)\) is established. We specify two subgraphs \(\Gamma_1(n)\) and \(\Gamma_2(n)\) of \(\Gamma(n)\). Let \(\Gamma_1(n)\) be induced by the vertices which are coprime to \(n\) and \(\Gamma_2(n)\) by induced by the set of vertices which are not coprime to \(n\). We determine the level of every component of \(\Gamma_1(n)\), and establish necessary and sufficient conditions when \(\Gamma_1(n)\) or \(\Gamma_2(n)\) has no cycles with length greater than \(1\), respectively. Moreover, the conditions for the semiregularity of \(\Gamma_2(n)\) are presented.

We made a computer search for minimal blocking sets in the projective geometry \(\text{PG}(2,11)\), and found \(30,000\), of which only two nontrivial blocking sets had the possibility of being isomorphic.

A graph \(G\) is called \({claw-free}\) if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). Ando et al. obtained the result: a claw-free graph \(G\) with minimum degree at least \(d\) has a path-factor such that the order of each path is at least \(d+1\); in particular \(G\) has a \(\{P_3, P_4, P_5\}\)-factor whenever \(d \geq 2\). Kawarabayashi et al. proved that every \(2\)-connected cubic graph has a \(\{P_3, P_4\}\)-factor. In this article, we show that if \(G\) is a connected claw-free graph with at least \(6\) vertices and minimum degree at least \(2\), then \(G\) has a \(\{P_3, P_4\}\)-factor. As an immediate consequence, it follows that every claw-free cubic graph (not necessarily connected) has a \(\{P_3, P_4\}\)-factor.

In this paper, we study the combinatorial properties of \(w\)-IPP (identifiable parents property) codes and give necessary and sufficient conditions for a code to be a \(w\)-IPP code. Furthermore, let \(R(C) = \frac{1}{n}{\log_q|C|}\) denote the rate of the \(q\)-ary code \(C\) of length \(n\), suppose \(q \geq 3\) is a prime power, we prove that there exists a sequence of linear \(q\)-ary \(2\)-IPP codes \(C_n\) of length \(n\) with \(R(C_n) = \frac{1}{3}log\frac{q^3}{4q^2-6q+3}\).

Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent, denoted \(G \sim H\), if \(P(G,\lambda) = P(H,\lambda)\). We write \([G] = \{H | H \sim G\}\). If \([G] = \{G\}\), then \(G\) is said to be chromatically unique. In this paper, we first characterize certain complete tripartite graphs \(G\) according to the number of \(4\)-independent partitions of \(G\). Using these results, we investigate the chromaticity of \(G\) with certain star or matching deleted. As a by-product, we obtain new families of chromatically unique complete tripartite graphs with certain star or matching deleted.

In this paper, we introduce a hyperoperation associated to the set of all arithmetic functions and analyze the properties of this new hyperoperation. Several characterization theorems are obtained, especially in connection with multiplicative functions.

In this paper, we introduce two new types of labelings of graphs using Fibonacci numbers, namely, Fibonacci graceful labelings and Super Fibonacci graceful labelings. We discuss the existence and non-existence of Fibonacci and Super Fibonacci graceful labelings for certain classes of graphs. Also, we discuss the Fibonacci gracefulness of disjoint union of Super Fibonacci graceful graphs, pendant edge extension of Super Fibonacci graceful graphs, and amalgamation of Super Fibonacci graceful graphs. Finally, we compare the graceful graphs with Fibonacci graceful graphs.

Let the columns of a \(p \times q\) matrix \(M\) over any ring be partitioned into \(n\) blocks, \(M = [M_1, \ldots, M_n]\). If no \(p \times p\) submatrix of \(M\) with columns from distinct blocks \(M_{i}\) is invertible, then there is an invertible \(p \times p\) matrix \(Q\) and a positive integer \(m \leq p\) such that \([QM_1, \ldots, QM_n]\) is in reduced echelon form and in all but at most \(m – 1\) blocks \(QM_i\) the last \(m\) entries of each column are either all zero or they include a non-zero non-unit.

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we determine the largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.