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 \( c \).

Given a labeling of the vertices and edges of a graph, we define a type of homogeneity that requires that the neighborhood of every vertex contains the same number of each of the labels. This homogeneity constraint is a generalization of regularity— all such graphs are regular. We consider a specific condition in which both the edge and vertex label sets have two elements and every neighborhood contains two of each label. We show that vertex homogeneity implies edge homogeneity (so long as the number of edges in any neighborhood is four), and give two theorems describing how to build new homogeneous graphs (or multigraphs) from others.

A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size \( 2 \) or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi_F'(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). A minimal \( F \)-coloring of \( G \) is an \( F \)-coloring with the property that if any red edge of \( G \) is re-colored blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the upper \( F \)-chromatic index \( \chi_F”(G) \) of \( G \). In this paper, we study the two color frames \( Y_1 \) and \( Y_2 \) that result from the claw \( K_{1,3} \), where \( Y_1 \) has exactly one red edge and \( Y_2 \) has exactly two red edges. For a graph \( G \), let \( \alpha'(G) \) and \( \alpha”(G) \) denote the matching number and lower matching number of \( G \), respectively. It is shown that if \( T \) is a tree of order at least \( 4 \) having no vertex of degree \( 2 \), then \( \chi_{Y_1}'(T) = \alpha”(T) \) while \( \chi_{Y_2}'(T) \leq 3\alpha”(T) \) and this upper bound is sharp. For a color frame \( F \) of a claw, sharp bounds are established for \( \chi_F”(G) \) in terms of the matching number and a generalized matching parameter of a graph \( G \). Other results and questions are also presented.

Let \( H \) and \( G \) be two simple graphs, where \( G \) is a subgraph of \( H \). A \( G \)-decomposition of \( \lambda H \), denoted by \( (\lambda H, G) \)-GD, is a partition of all the edges of \( \lambda H \) into subgraphs (G-blocks), each of which is isomorphic to \( G \). A large set of \( (\lambda H, G) \)-GD, denoted by \( (\lambda H, G) \)-LGD, is a partition of all subgraphs isomorphic to \( G \) of \( H \) into \( (\lambda H, G) \)-GDs (called small sets). In this paper, we investigate the existence of \( (\lambda K_{mn}, K_{1,p}) \)-LGD and obtain some existence results, where \( p \geq 3 \) is a prime.

Let \( M(b, n) \) be the complete multipartite graph with \( b \) parts \( B_0, \dots, B_{b-1} \) of size \( n \). A \( z \)-cycle system of \( M(b, n) \) is said to be a \({cycle-frame}\) if the \( z \)-cycles can be partitioned into sets \( S_1, \dots, S_k \) such that for \( 1 \leq j \leq k \), \( S_j \) induces a \( 2 \)-factor of \( M(b, n) \backslash B_i \) for some \( i \in \mathbb{Z}_b \). The existence of a \( C_z \)-frame of \( M(b, n) \) has been settled when \( z \in \{3, 4, 5, 6\} \). Here, we completely settle the case of \( C_z \)-frames when \( z \) is \( 8 \), and we give some solutions for larger values of \( z \).

A graph \( G \) is said to be a \( (2, k) \)-regular graph if each vertex of \( G \) is at a distance two away from \( k \) vertices of \( G \). A graph \( G \) is called an \( (r, 2, k) \)-regular graph if each vertex of \( G \) is at a distance \( 1 \) away from \( r \) vertices of \( G \) and each vertex of \( G \) is at a distance \( 2 \) away from \( k \) vertices of \( G \) \cite{9}. This paper suggests a method to construct a \( ((m + n – 2), 2, (m – 1)(n – 1)) \)-regular graph of smallest order \( mn \) containing a given graph \( G \) of order \( n \geq 2 \) as an induced subgraph for any \( m > 1 \).

A broadcast on a graph \( G \) is a function \( f: V \to \{0, 1, \dots, \text{diam}G\} \) such that \( f(v) < e(v) \) (the eccentricity of \( v \)) for all \( v \in V \). The broadcast number of \( G \) is the minimum value of \( \sum_{v \in V} f(v) \) among all broadcasts \( f \) for which each vertex of \( G \) is within distance \( f(v) \) from some vertex \( v \) with \( f(v) \geq 1 \). This number is bounded above by the radius of \( G \). A graph is uniquely radial if its only minimum broadcasts are broadcasts \( f \) such that \( f(v) = \text{rad}G \) for some central vertex \( v \), and \( f(u) = 0 \) if \( u \neq v \). We characterize uniquely radial trees.