A vertex \( u \) in a set \( S \) of vertices of a graph \( G \) has a private neighbor (with respect to \( S \)) if either (i) \( u \) has no neighbor in \( S \) or (ii) there exists some vertex \( v \in V(G) – S \) that is a neighbor of \( u \) but not a neighbor of any other vertex in \( S \). A set \( S \) is irredundant in a graph \( G \) if every vertex in \( S \) has a private neighbor. An irredundant \( k \)-coloring of a graph \( G \) is a partition of \( V(G) \) into \( k \) irredundant sets. The minimum \( k \) for which \( G \) has an irredundant \( k \)-coloring is the irredundant chromatic number \( \chi_{ir}(G) \) of \( G \).

A complete coloring of a graph \( G \) is a proper vertex coloring of \( G \) having the property that for every two distinct colors \( i \) and \( j \) used in the coloring, there exist adjacent vertices of \( G \) colored \( i \) and \( j \). The maximum positive integer \( k \) for which \( G \) has a complete \( k \)-coloring is the achromatic number \( \psi(G) \) of \( G \).

A Grundy coloring of a graph \( G \) is a proper vertex coloring of \( G \) having the property that for every two colors (positive integers) \( i \) and \( j \) with \( i < j \), every vertex colored \( j \) has a neighbor colored \( i \). The maximum positive integer \( k \) for which a graph \( G \) has a Grundy \( k \)-coloring is the Grundy number \( \Gamma(G) \) of \( G \). The chromatic number of a graph \( G \) is denoted by \( \chi(G) \). For every graph \( G \), these four coloring parameters satisfy the inequalities \( \chi_{ir}(G) \leq \chi(G) \leq \Gamma(G) \leq \psi(G) \). It is shown that if \( a, b, c \), and \( d \) are integers with \( 2 \leq a \leq b \leq c \leq d \), then there exists a nontrivial connected graph \( G \) with \( \chi_{ir}(G) = a \), \( \chi(G) = b \), \( \Gamma(G) = c \), and \( \psi(G) = d \) if and only if \( d = 2 \) or \( c \neq 2 \).

Let \(\Sigma\) be a totally ordered set. We work on finite strings \(b = b_1 b_2 \ldots b_m\) of \(b_i\) elements from \(\Sigma\). Such a \(b\) is a Lyndon word (Lyn) if \(m \geq 1\), and \(b\) is the unique first in lexicographic order among the \(m\) rows of the \(m \times m\) circulant matrix with \(b\) as the first row.

A classic result is that every string \(b\) has a unique maximal factorization \(umf(b)\) into Lyndon words, each Lyndon word of the maximum possible size in \(b\).

In 1983, J. P. Duval \([6]\) published Algorithm 1, which finds \(umf(b)\). It was studied in 1991 by A. Apostolico and M. Crochemore \([1]\). Their work was then studied in 1994 by J.W. Daykin, C.S. Iliopoulos, and W.F. Smyth \([5]\).

Since Duval used a programming language, we start by giving a new simple account of his Algorithm 1. Our Algorithm 2 modifies Duval’s Algorithm 1 to find \(umf(a)\), when \(a\) is a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_i\).

Our Algorithm 3 is also for a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_i\). It is completely different from Algorithms 1 and 2. It snakes right, left, right, and so on. It revealed that Lyndon words have a special structure. We give an example where Algorithm 3 needs almost \(2m\) tests; we think that is the most needed, but cannot give a rigorous proof.

Let \(\Sigma\) be a totally ordered set. We work on finite strings \(b = b_1 b_2 \ldots b_m\) of \(b_i\) elements from \(\Sigma\). Such a \(b\) is a lyn(Lyndon word) if \(m \geq 1\), and \(b\) is the unique first in lex(lexicographic order) among the \(m\) rows of the \(m \times m\) circulant matrix with \(b\) as first row.

A classic result is that every string \(b\) has a unique max factorization \(umf(b)\) into Lyns , each Lyn of maximum possible size in \(b\).

In 1983 J. P. Duval [6] published Algorithm 1, which finds \(umf(b)\). It was studied in 1991 by A. Apostolico and M. Crochemore [1]. Then their work was studied in 1994 by J.W. Daykin, C.S. Iliopoulos, and W.F. Smyth [5].

Since Duval used a programming language, we start by giving a new simple account of his Algorithm 1. Then our Algorithm 2 given here modifies Duval’s Algorithm 1 to find \(umf(a)\), when \(a\) is a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_I\).

Our Algorithm 3 is also for a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_I\). It is completely different from Algorithms 1 and 2. It snakes right, left, right, and so on. It revealed the fact that Lyndon words have a special structure. We give an example where Algorithm 3 needs almost \(2m\) tests; we think that is the most needed, but cannot give a rigorous proof.

We find interesting properties of Lyns, some of which may be new.

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\dim(G)\) is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\).

The metric dimension of some classes of plane graphs has been determined in \([3]\), \([4]\), \([5]\), \([10]\), \([13]\), and \([18]\), while the metric dimension of some classes of convex polytopes has been determined in \([8]\), and a question was raised as an open problem: Is it the case that the graph of every convex polytope has constant metric dimension? In this paper, we study the metric dimension of two classes of convex polytopes. It is shown that these classes of convex polytopes have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

The degree set of a graph \( G \) is the set \( S \) consisting of the distinct degrees of vertices in \( G \). In 1977, Kapoor, Polimeni, and Wall \([2]\) determined the least number of vertices among simple graphs with a given degree set. In this note, we look at the analogue problem concerning the least order and the least size of a multigraph with a given degree set.

Let \(\mathcal{P}\) be a graph property and \(G\) a graph. \(G\) is said to be \(\mathcal{P}\)-saturated if \(G\) does not have property \(\mathcal{P}\) but the addition of any edge between non-adjacent vertices of \(G\) results in a graph with property \(\mathcal{P}\). If \(\mathcal{P}\) is a bipartite graph property and \(G\) is a bipartite graph not in \(\mathcal{P}\), but the addition of any edge between non-adjacent vertices in different parts results in a graph in \(\mathcal{P}\), then \(G\) is \(\mathcal{P}\)-bisaturated. We characterize all \(\mathcal{P}\)-saturated graphs, for which \(\mathcal{P}\) is the family of interval graphs, and show that this family is precisely the family of maximally non-chordal graphs. We also present a conjectured characterization of all \(\mathcal{P}\)-bisaturated graphs, in the case where \(\mathcal{P}\) is the family of interval bigraphs, and prove it as far as current forbidden subgraph characterizations allow. We demonstrate that extremal non-interval graphs and extremal non-interval bigraphs are highly related, in that the former is simply a complete graph with \(2K_2\) removed and the latter is a complete bipartite graph with \(3K_2\) removed.

The Stein-Lovasz Theorem can be used to get existence results for some combinatorial problems using constructive methods rather than probabilistic methods. In this paper, we discuss applications of the Stein-Lovasz Theorem to some combinatorial set systems and arrays, including perfect hash families, separating hash families, splitting systems, covering designs, lotto designs and \( A \)-free systems. We also compare some of the bounds obtained from the Stein-Lovasz Theorem to those using the basic probabilistic method.

The distribution of distances in the star graph \( S_{T_n} \) (\(1 < n \in \mathbb{Z}\)) is established, and subsequently a threaded binary tree is obtained that realizes an orientation of \( S_{T_n} \) whose levels are given by the distances to the identity permutation, via a pruning algorithm followed by a threading algorithm. In the process, the distributions of distances of the efficient dominating sets of \( S_{T_n} \) are determined.