A primitive hypergraph is a hypergraph with maximum cardinality three and maximum degree three such that every \(3\)-edge is adjacent only to \(2\)-edges and is incident only to vertices of degree two. Deciding the bicolorability of a primitive hypergraph is NP-complete (a straightforward consequence of results in [14]). We provide sufficient conditions, similar to the Sterboul conditions proved by Défossez [5], for the existence of a bicoloring of a primitive hypergraph, and we provide a polynomial algorithm for bicoloring a primitive hypergraph if those conditions hold. We then draw a connection between this algorithm and the well-known necessary and sufficient conditions given by Berge [1] for maximal matchings in graphs, which leads to a characterization of bicolorability of primitive hypergraphs.

Let \( D \) be a strongly connected oriented graph with vertex-set \( V \) and arc-set \( A \). The distance from a vertex \( u \) to another vertex \( v \), \( d(u,v) \), is the minimum length of oriented paths from \( u \) to \( v \). Suppose \( B = \{b_1, b_2, b_3, \ldots, b_k\} \) is a nonempty ordered subset of \( V \). The representation of a vertex \( v \) with respect to \( B \), \( r(v|B) \), is defined as a vector \( (d(v,b_1), d(v,b_2), \ldots, d(v,b_k)) \). If any two distinct vertices \( u,v \) satisfy \( r(u|B) \neq r(v|B) \), then \( B \) is said to be a resolving set of \( D \). If the cardinality of \( B \) is minimum, then \( B \) is said to be a basis of \( D \), and the cardinality of \( B \) is called the directed metric dimension of \( D \).

Let \( G \) be the underlying graph of \( D \) admitting a \( C_n \)-covering. A \( C_n \)-simple orientation is an orientation on \( G \) such that every \( C_n \) in \( D \) is strongly connected. This paper deals with metric dimensions of oriented wheels, oriented fans, and amalgamation of oriented cycles, all of which admit \( C_n \)-simple orientations.

A Stanton-type graph \( S(n, m) \) is a connected multigraph on \( n \) vertices such that for a fixed integer \( m \) with \( n – 1 \leq m \leq \binom{n}{2} \), there is exactly one edge of multiplicity \( i \) (and no others) for each \( i = 1, 2, \ldots, m \). In a recent paper, the authors decomposed \( \lambda K_{n} \) (for the appropriate minimal values of \( \lambda \)) into two of the four possible types of \( S(4, 3) \)’s. In this note, decompositions of \( \lambda K_{n} \) (for the appropriate minimal values of \( \lambda \)) into the remaining two types of \( S(4, 3) \)’s are given.

In this paper, we describe a backtrack search over parallel classes with a partial isomorph rejection to classify resolvable \(2\)-(12, 6, \(5c\)) designs. We use the intersection pattern between the parallel classes and the fact that any resolvable \(2\)-(12, 6, \(5c\)) design is also a resolvable \(3\)-(12, 6, \(2c\)) design to effectively guide the search. The method was able to enumerate all nonsimple resolutions and a subfamily of simple resolutions of a \(2\)-(12, 6, 15) design. The method is also used to confirm the computer classification of the resolvable \(2\)-(12, 6, \(5c\)) designs for \(c \in \{1, 2\}\). A consistency checking based on the principle of double counting is used to verify the computation results.

A restraint on a (finite undirected) graph \( G = (V, E) \) is a function \( r \) on \( V \) such that \( r(v) \) is a finite subset of \( \mathbb{N} \); a proper vertex colouring \( c \) of \( G \) is permitted by \( r \) if \( c(v) \notin r(v) \) for all vertices \( v \) of \( G \) (we think of \( r(v) \) as the set of colours forbidden at \( v \)). Given a large number of colors, for restraints \( r \) with exactly one colour forbidden at each vertex the smallest number of colourings is permitted when \( r \) is a constant function, but the problem of what restraints permit the largest number of colourings is more difficult. We determine such extremal restraints for complete graphs and trees.

Let \( G \) be a graph with average degree greater than \( k – 2 \). Erdős and Sós conjectured that \( G \) contains every tree on \( k \) vertices. A star is a tree consisting of one center vertex adjacent to all the other vertices, and a \({double-broom}\) is a tree made up of two stars and a path connecting the center of one star with the center of the other. If the path connecting the two stars has length 2 or 3, then \( G \) contains the double-broom (unpublished). In this paper, we prove that \( G \) contains every double-broom on \( k \) vertices.

We indicate how to calculate the number of round-robin tournaments realizing a given score sequence. This is obtained by inductively calculating the number of tournaments realizing a score function. Tables up to 18 participants are obtained.

An urn contains \(2n + 1\) balls in two colors. The number of balls of a particular color is a random variable having binomial distribution with \( p = \frac{1}{2} \). We sample the urn removing balls one by one without replacement. Our aim is to stop the process maximizing the probability that the color of the last selected ball is the minority color. We give an algorithm for an optimal stopping time, evaluate the probability of success and its asymptotic behavior.

The results of Laughlin and Johnson [1] are generalized in this paper, and open problems left at the end of [1] are addressed. New values of Anti-Waring numbers are given, including \( N(2,4) \), \( N(2,5) \), \( N(2,6) \), and \( N(2,7) \).

A function \( f: V(G) \to \{0, 1, 2\} \) is a \({Roman\; dominating\; function}\) (or just RDF) if every vertex \( u \) for which \( f(u) = 0 \) is adjacent to at least one vertex \( v \) for which \( f(v) = 2 \). The weight of a Roman dominating function is the value \( f(V(G)) = \sum_{u \in V(G)} f(u) \). The \({Roman\; domination\; number}\) of a graph \( G \), denoted by \( \gamma_R(G) \), is the minimum weight of a Roman dominating function on \( G \). A graph \( G \) is Roman domination critical upon edge subdivision if the Roman domination number increases whenever an edge is subdivided. In this paper, we study the Roman domination critical graphs upon edge subdivision. We present several properties, bounds, and general results for these graphs.