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 If the path connecting the two stars has length 2 or 3, then \( G \) contains the double-broom (unpublished).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.

In [Discrete Math., 311 (2011), 688-689], Fujita defined \( f(r,n) \) to be the maximum integer \( k \) such that every \( r \)-edge-coloring of \( K_n \) contains a monochromatic cycle of length at least \( k \). In this paper, we investigate the values of \( f(r,n) \) when \( n \) is linear in \( r \). We determine the value of \( f(r, 2r+2) \) for all \( r \geq 1 \) and show that \( f(r, sr+c) = s+1 \) if \( r \) is sufficiently large compared with positive integers \( s \) and \( c \).