If \(f\) and \(g\) are self-maps on a finite set \(M\) with \(n = |M|\), then the images of various composite functions such as \(f^2gf\) and \(g^2 f^2 g\) may have different sizes. There is, of course, a minimal image size which can be achieved by the composition of particular functions. It can be difficult, however, to discover the size of this minimal image. We seek to determine “words” over a finite alphabet \(S \) which, by specifying function compositions when letters are interpreted as functions, allow one to test for each \(k\) whether or not there exists among all compositions an image of size \(n – k\) or less. For two functions \(f\) and \(g\), \(W_1 = fg\) is clearly such a “word” for \(k = 1\), since no composition of functions \(f\) and \(g\) has an image smaller than or equal to \(|M| – 1\), if \(W_1 = fg\) fails to do so. We prove the existence of such a word \(W_k\) for each \(k\), and exhibit a recursive procedure for the generation of \(W_{k+1}\) from \(W_k\). The words \(W_k\) depend only upon the finite alphabet \( S \), and are independent of the size of the finite set \(M\) over which the symbols from \( S \) are to be interpreted as functions.

It is shown that a symmetric design with \(\lambda=2\) can admit \(PSL(2,q)\) for \(q\) odd and \(q\) greater than \(3\) as an automorphism group fixing a block and acting in its usual permutation representation on the points of the block only if \(q\) is congruent to \(5\pmod{8}\). A consequence for more general automorphism groups is also described.

In this paper, we consider the structure of \(k\)-saturated graphs \((G \not\supset K_k,\) but \(G+e \supset K_{k}\) for all possible edges \(e)\\) having chromatic number at least \(k\).

In this paper, the authors study the vulnerability parameters of integrity, toughness, and binding number for two classes of graphs. These two classes of graphs are permutation graphs of complete graphs and permutation graphs of complete bipartite graphs

In this paper we examine bounds on \(|N(x) \cup N(y)|\) (for nonadjacent pairs \(x,y \in V(G)\)) that imply certain strong Hamiltonian properties in graphs. In particular, we show that if \(G\) is a 2-connected graph of order \(n\) and if for all pairs of distinct nonadjacent vertices \(x, y \in V(G)\),

1. \(|N(z) \cup N(y)| \geq \frac{2n+5}{3}\), then \(G\) is pancyclic.
2. \(|N(z) \cup N(y)| \geq n-t\) and \(\delta(G) \geq t\), then \(G\) is Hamiltonian.
3. \(|N(z) \cup N(y)| \geq n-2\), then \(G\) is vertex pancyclic.

Three types of graphs are investigated with respect to cordiality, namely:graphs which are the complete product of two cordial graphs, graphs which are the subdivision graphs of cordial graphs, cactus graphs.
We give sufficient conditions for the cordiality of graphs of the first two types and show that a cactus graph is cordial if and only if the cardinality of its edge set is not congruent to \(2\) (mod 4).

It is shown that there exists a 4-critical 3-uniform linear hypergraph of order \(m\) for every \(m \geq 56\).

Essentially all pairs of forests \((F_1,F_2)\) are determined for which \(R(F_1,F_2)\) is finite, where \(R(F_1,F_2)\) is the class of minimal Ramsey graphs for the pair \((F_1,F_2)\).

Steiner Heptagon Systems (SHS) of type 1, 2, and 3 are defined and the spectrum of type 2 SHSs (SHS2) is studied. It is shown that the condition \(n \equiv 1 \) { or } \(7 \pmod{14}\) is not only necessary but also sufficient for the existence of an SHS2 of order \(n\), with the possible exceptions of \(n=21\) and \(85\). This gives an interesting algebraic result since the study of SHS2s is equivalent to the study of quasigroups satisfying the identities \(x^2 = x\), \((yx)x = y\), and \((xy)(y(xy)) = (yx)(x(yx))\).