A multiple shell \(MS\{n_1^{t_1}, n_2^{t_2}, \dots, n_r^{t_r}\}\) is a graph formed by \(t_i\) shells of widths \(n_i\), \(1 \leq i \leq r\), which have a common apex. This graph has \(\sum_{i=1}^rt_i(n_i-1) + 1\) vertices. A multiple shell is said to be balanced with width \(w\) if it is of the form \(MS\{w^t\}\) or \(MS\{w^t, (w+1)^s\}\). Deb and Limaye have conjectured that all multiple shells are harmonious, and shown that the conjecture is true for the balanced double shells and balanced triple shells. In this paper, the conjecture is proved to be true for the balanced quadruple shells.

In [BabStein], Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In \([Kit1]\), Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There, either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper, we find the exponential generating function for the remaining case. Also, we consider permutations that avoid a pattern of the form \(x – yz\) or \(xy – z\) and begin with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(23\ldots 1k\), \((k-1)(k-2)\ldots 1k\), or end with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(1k(k-1)\ldots 2\), \(k12\ldots (k-1)\). For each of these cases, we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides, we generalize some of the obtained results as well as some of the results given in \([Kit3]\): we consider permutations avoiding certain generalized \(3\)-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.

In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (\(LEE\)) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of \(LEE\)-graphs over the set of all connected spanning \(k\)-edge subgraphs of a connected graph \(G\), and show that both of them are connected for a \(2\)-connected graph \(G\).

We look at binary strings of length \(n\) which contain no odd run of zeros and express the total number of such strings, the number of zeros, the number of ones, the total number of runs, and the number of levels, rises, and drops as functions of the Fibonacci and Lucas numbers and also give their generating functions. Furthermore, we look at the decimal value of the sum of all binary strings of length \(n\) without odd runs of zeros considered as base \(2\) representations of decimal numbers, which interestingly enough are congruent (mod \(3\)) to either \(0\) or a particular Fibonacci number. We investigate the same questions for palindromic binary strings with no odd runs of zeros and obtain similar results, which generally have different forms for odd and even values of \(n\).

In this paper, we characterize the potentially \(K_4\)-graphic sequences. This characterization implies the value \(\sigma(K_4,n)\), which was conjectured by P. Erdős, M. S. Jacobson, and J. Lehel [1] and was confirmed by R. J. Gould, M. S. Jacobson, and J. Lehel [2] and Jiong-Sheng Li and Zixia Song [5], independently.

The graph …….is called a kite and the decomposition of \(K_n\) into kites is called a kite system. Such systems exist precisely when \(n = 0\) or \(1\) (mod \(8\)). In \(1975\), C. C. Lindner and A. Rosa solved the intersection problem for Steiner triple systems. The object of this paper is to give a complete solution to the triangle intersection problem for kite systems (\(=\) how many triangles can two kite systems of order \(n\) have in common). We show that if \(x \in \{0, 1, 2, \dots, n(n-1)/8\}\), then there exists a pair of kite systems of order \(n\) having exactly \(n\) triangles in common.

A directed balanced incomplete block design (\(DB\)(\(k\), \(\lambda\);\(v\))) \((X, \mathcal{B})\) is called self-converse if there is an isomorphic mapping \(f\) from \((X, \mathcal{B})\) to \((X, \mathcal{B}^{-1})\), where \(\mathcal{B}^{-1} = \{B^{-1} : B \in \mathcal{B}\}\) and \(B^{-1} = (x_k,x_{k-1},\ldots,x_2,x_1)\) for \(B=(x_1,x_2,\ldots,x_{k-1},x_{k})\). In this paper, we give the existence spectrum for self-converse \(DB\)(\(4\),\(\lambda\);\(v\)) for any \(\lambda \geq 1\).

A sequence \(\pi = (d_1, \dots, d_n)\) of nonnegative integers is graphic if there exists a graph \(G\) with \(n\) vertices for which \(d_1, \dots, d_n\) are the degrees of its vertices. \(G\) is referred to as a realization of \(\pi\). Let \(P\) be a graph property. A graphic sequence \(\pi\) is potentially \(P\)-graphic if there exists a realization of \(\pi\) with the graph property \(P\). Similarly, \(\pi\) is forcibly \(P\)-graphic if all realizations of \(\pi\) have the property \(P\). We characterize potentially Halin graph-graphic sequences, forcibly Halin graph-graphic sequences, and forcibly cograph-graphic sequences.

We establish the nonexistence of:(i) Steiner \(t\)-\((v,k)\) trades of volume \(s\), for \(2^t + 2^{t-1} < s t+1\) and volume \(s < (t-1)2^t + 2\).