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.