We classify all finite linear spaces on at most \(15\) points admitting a blocking set. There are no such spaces on \(11\) or fewer points, one on \(12\) points, one on \(13\) points, two on \(14\) points, and five on \(15\) points. The proof makes extensive use of the notion of the weight of a point in a \(2\)-coloured finite linear space, as well as the distinction between minimal and non-minimal \(2\)-coloured finite linear spaces. We then use this classification to draw some conclusions on two open problems on the \(2\)-colouring of configurations of points.

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.

For a group \(T\) and a subset \(S\) of \(T\), the bi-Cayley graph \(\text{BCay}(T, S)\) of \(T\) with respect to \(S\) is the bipartite graph with vertex set \(T \times \{0, 1\}\) and edge set \(\{\{(g, 0), (ag, 1)\} | g \in T, s \in S\}\). In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infinite families of cubic semisymmetric graphs.

We study the notion of path-congruence \(\Phi: T_1 \rightarrow T_2\) between two trees \(T_1\) and \(T_2\). We introduce the concept of the trunk of a tree, and prove that, for any tree \(T\), the trunk and the periphery of \(T\) are stable. We then give conditions for which the center of \(T\) is stable. One such condition is that the central vertices have degree \(2\). Also, the center is stable when the diameter of \(T\) is less than \(8\).

We call a cycle whose length is at most \(5\) a short cycle. In this paper, we consider the packing of short cycles in a graph with specified edges. A minimum degree condition is obtained, which is slightly weaker than that of the result in \([1]\).

Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \(f\)-factor if \(d_G^{h}(x) = f(x)\) for every \(x \in V(F)\). In this paper, we prove that if \(\delta(G) \geq b\) and \(\alpha(G) \leq \frac{4a(\delta-b)}{(b+1)^2}\), then \(G\) has a fractional \(f\)-factor. Where \(a\) and \(b\) are integers such that \(0 \leq a \leq f(x) \leq b\) for every \(x \in V(G)\). Therefore, we prove that the fractional analogue of Conjecture in \([2]\) is true.

Let \(D\) be a connected symmetric digraph, \(A\) a finite abelian group, \(g \in A\) and \(\Gamma\) a group of automorphisms of \(D\). We consider the number of \(T\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order. Specifically, we enumerate the number of \(I\)-isomorphism classes of connected \(g\)-cyclic \(A\)-covers of \(D\) for an element \(g\) of odd order and the trivial automorphism group \(\Gamma\) of \(D\), when \(A\) is the cyclic group \({Z}_{p^n}\) and the direct sum of \(m\) copies of \({Z}_p\) for any prime number \(p (> 2)\).

The Grundy number of an impartial game \(G\) is the size of the unique Nim heap equal to \(G\). We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these “fractal sequences” and give a bijection between these sequences and certain upper-triangular arrays. As a special case, we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap.

A graph \(G\) is clique-perfect if the cardinality of a maximum clique-independent set of \(H\) is equal to the cardinality of a minimum clique-transversal of \(H\), for every induced subgraph \(H\) of \(G\). When equality holds for every clique subgraph of \(G\), the graph is \(c\)-clique-perfect. A graph \(G\) is \(K\)-perfect when its clique graph \(K(G)\) is perfect. In this work, relations are described among the classes of perfect, \(K\)-perfect, clique-perfect and \(c\)-clique-perfect graphs. Besides, partial characterizations of \(K\)-perfect graphs using polyhedral theory and clique subgraphs are formulated.

In this note, we investigate arithmetic properties of the Motzkin numbers. We prove that for large \(n\), the product of the first \(n\) Motzkin numbers is divisible by a large prime. The proofs use the Deep Subspace Theorem.

The point-distinguishing chromatic index of a graph \(G\), denoted by \(\chi_o(G)\), is the smallest number of colours in a (not necessarily proper) edge colouring of \(G\) such that any two distinct vertices of \(G\) are distinguished by sets of colours of their adjacent edges. The exact value of \(\chi_o(K_{m,n})\) is found if either \(m \leq 10\) or \(n \geq 8m^2 – 2m + 1\).

Star graphs were introduced by \([1]\) as a competitive model to the \(n\)-cubes. Then hyper-stars were introduced in \([9]\) to be a competitive model to both \(n\)-cubes and star graphs. In this paper, we discuss strong connectivity properties and orientability of the hyper-stars.

In this paper, three methods for constructing larger harmonious graphs from one or a set of harmonious graphs are provided.

The complexity of determining if a Steiner triple system on \(v = 6n + 3\) points contains a parallel class is currently unknown. In this paper, we show that the problem of determining if a partial Steiner triple system on \(v = 6n + 3\) points contains a parallel class is NP-complete. We also consider the problem of determining the chromatic index of a partial Steiner triple system and show that this problem is NP-hard.

In this paper, it has been proved that \(K_{r,r} \times K_{m}\), \(m \geq 3\), is hamiltonian decomposable.

A twofold extended triple system with two idempotent elements, \(TETS(v)\), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of triples, called blocks, of type \(\{x,y,z\}\), \(\{x,x,y\}\) or \(\{x,x,x\}\) such that every pair of elements of \(V\), not necessarily distinct, belongs to exactly two triples and there are only two triples of the type \(\{x, x, x\}\).
This paper shows that an indecomposable \(TETS(v)\) exists which contains exactly \(k\) pairs of repeated blocks if and only if \(v \not\equiv 0 \mod 3\), \(v \geq 5\) and \(0 \leq k \leq b_v – 2\), where \(b_v = \frac{(v + 2)(v + 1)}{6}\).

For a subset of vertices \(S\) in a graph \(G\), if \(v \in S\) and \(w \in V-S\), then the vertex \(w\) is an \(external\; private\; neighbor\; of \;v\) (with respect to \(S\)) if the only neighbor of \(w\) in \(S\) is \(v\). A dominating set \(S\) is a private dominating set if each \(v \in S\) has an external private neighbor. Bollébas and Cockayne (Graph theoretic parameters concerning domination, independence and irredundance. J. Graph Theory \(3 (1979) 241-250)\) showed that every graph without isolated vertices has a minimum dominating set which is also a private dominating set. We define a graph \(G\) to be a \(private\; domination\; graph\) if every minimum dominating set of \(G\) is a private dominating set. We give a constructive characterization of private domination trees.

The Levi graph of a balanced incomplete block design is the bipartite graph whose vertices are the points and blocks of the design, with each block adjacent to those points it contains. We derive upper and lower bounds on the isoperimetric numbers of such graphs, with particular attention to the special cases of finite projective planes and Hadamard designs.