Suppose \(G\) is a finite plane graph with vertex set \(V(G)\), edge set \(E(G)\), and face set \(F(G)\). The paper deals with the problem of labeling the vertices, edges, and faces of a plane graph \(G\) in such a way that the label of a face and labels of vertices and edges surrounding that face add up to a weight of that face. A labeling of a plane graph \(G\) is called \(d\)-antimagic if for every number \(s\), the \(s\)-sided face weights form an arithmetic progression of difference \(d\). In this paper, we investigate the existence of \(d\)-antimagic labelings for a special class of plane graphs.

The choice number of a graph \(G\), denoted by \(\chi_l(G)\), is the minimum number \(\chi_l\) such that if we give lists of \(\chi_l\) colors to each vertex of \(G\), there is a vertex coloring of \(G\) where each vertex receives a color from its own list no matter what the lists are. In this paper, we show that \(\chi_l(G) \leq 3\) for each plane graph of girth at least \(4\) which contains no \(8\)-circuits and \(9\)-circuits.

It is noted that Teirlinck’s “transposition argument” for disjoint \(\text{STS}(v)\) applies more generally to certain partial triple systems of different orders. A corollary on the number of blocks common to two \(\text{STS}(v)\) of different orders is also given.

We introduce a generalisation of the traditional magic square, which proves useful in the construction of magic labelings of graphs. An order \(n\) sparse semi-magic square is an \(n \times n\) array containing the entries \(1, 2, \ldots, m\) (for some \(m < n^2\)) once each with the remainder of its entries \(0\), and its rows and columns have a constant sum \(k\). We discover some of the basic properties of such arrays and provide constructions for squares of all orders \(n \geq 3\). We also show how these arrays can be used to produce vertex-magic labelings for certain families of graphs.

A graph \(G\) on \(n\) vertices has a prime labeling if its vertices can be assigned the distinct labels \(1, 2, \ldots, n\) such that for every edge \(xy\) in \(G\), the labels of \(x\) and \(y\) are relatively prime. In this paper, we show that generalized books and \(C_m\) snakes all have prime labelings. In the process, we demonstrate a way to build new prime graphs from old ones.

In this paper, we studied that a linear space, which is the complement of a linear space having points are not on a trilateral or a quadrilateral in a projective subplane of order \(m\), is embeddable in a unique way in a projective plane of order \(n\). In addition, we showed that this linear space is the complement of certain regular hyperbolic plane in the sense of Graves \([5]\) with respect to a finite projective plane.

We give a combinatorial proof of Wilson’s Theorem: \(p\) divides \(\{(p – 1)! +1\}\) if \(p\) is prime.

The Padmakar-Ivan (PI) index of a graph \(G\) is defined as \(PI(G) = \sum[n_{eu} (e|G) + n_{ev}(e|G)]\) where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\) than to \(v\), \(n_{ev}(e|G)\) is the number of edges of \(G\) lying closer to \(v\) than to \(u\), and the summation goes over all edges of \(G\). The PI Index is a Szeged-like topological index developed very recently. In this paper, an exact expression for the PI index of the armchair polyhex nanotubes is given.

A finite planar set is \(k\)-isosceles for \(k \geq 3\), if every \(k\)-point subset of the set contains a point equidistant from the other two. This paper gives a \(4\)-isosceles set consisting of \(7\) points with no three on a line and no four on a circle.