It is shown that for \( n \geq 16 \), the sum of cardinalities of open irredundant sets in an \( n \)-vertex graph and its complement is at most \( \frac{3n}{4} \).

The redundance \( R(G) \) of a graph \( G \) is the minimum, over all dominating sets \( S \), of \( \sum_{v \in S} (1 + \deg(v)) \), where \( \deg(v) \) is the degree of vertex \( v \). We use some dynamic programming algorithms to compute the redundance of complete grid graphs \( G_{m,n} \) for \( 1 \leq m \leq 21 \) and all \( n \), and to establish good upper and lower bounds on the redundance for larger \( m \). We conjecture that the upper bound is the redundance when \( m > 21 \).

Heinrich et al. [4] characterized those simple eulerian graphs with no Petersen-minor which admit a triangle-free cycle decomposition, a TFCD. If one permits Petersen minors then no such characterization is known even for \( {E}(4,2) \), the set of all the eulerian graphs of maximum degree 4. Let \( {EM}(4,2) \subset {E}(4,2) \) be the set of all graphs \( H \) such that all triangles of \( H \) are vertex disjoint, and each triangle contains a degree 2 vertex in \( H \). In the paper it is shown that to each \( G \in {E}(4,2) \) there exists a finite subset \( S \subset {EM}(4,2) \) so that \( G \) admits a TFCD if and only if some \( H \in S \) admits a TFCD. Further, some sufficient conditions for a graph \( G \in {E}(4,2) \) to possess a TFCD are given.

Let \( \nu \) be some graph parameter and let \( \mathcal{G} \) be a class of graphs for which \( \nu \) can be computed in polynomial time. In this situation, it is often possible to devise a strategy to decide in polynomial time whether \( \nu \) has a unique realization for some graph in \( \mathcal{G} \). We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for three well-known graph parameters: the domination number, the independence number, and the chromatic number.

A \( k \)-line-distinguishing coloring of a graph \( G = (V, E) \) is a partition of \( V \) into \( k \) sets \( V_1, \ldots, V_k \) such that \( q(\langle V_i \rangle) \leq 1 \) for \( i = 1, \ldots, k \) and \( q(V_i, V_j) \leq 1 \) for \( 1 \leq i \leq j \leq k \). If the color classes in a line-distinguishing coloring are also independent, then it is called a harmonious coloring. A coloring is minimal if, when two color classes are combined, we no longer have a coloring of the given type.

The upper harmonious chromatic number, \( H(G) \), is defined as the maximum cardinality of a minimal harmonious coloring of a graph \( G \), while the upper line-distinguishing chromatic number, \( H'(G) \), is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \( G \). For any graph \( G \) of maximum degree \( \Delta(G) \), \( H'(G) \geq \Delta(G) \) and \( H(G) \geq \Delta(G) + 1 \).

We characterize connected graphs \( G \) that contain neither a triangle nor a 5-cycle for which \( H(G) = \Delta(G) + 1 \). We show that a triangle-free connected graph \( G \) satisfies \( H'(G) = \Delta(G) \) if and only if \( G \) is a star \( K_{1, \Delta(G)} \). A partial characterization of connected graphs \( G \) for which \( H'(G) = \Delta(G) \) is obtained.

There are at least 52432 symmetric \( (100, 45, 20) \) designs on which \( \text{Frob}_{10} \times \mathbb{Z}_2 \) acts as an automorphism group. All these designs correspond to Bush-type Hadamard matrices of order 100, and each leads to an infinite class of twin designs with parameters
\[v= 100(81^m + 81^{m-1} + \ldots + 81+1),\, k=45(81)^m ,\, \lambda=20(81)^m ,\]
and an infinite class of Siamese twin designs with parameters
\[v= 100(121^m + 121^{m-1} + \ldots + 121+1),\, k=55(121)^m ,\, \lambda=30(121)^m ,\]
where \( m \) is an arbitrary positive integer. One of the constructed designs is isomorphic to that used by Z. Janko, H. Kharaghani, and V. D. Tonchev [4].

We define the \( B_2 \) block-intersection graph of a balanced incomplete block design \( (V,\mathfrak{B}) \) having order \( n \), block size \( k \), and index \( \lambda \), or BIBD\( (n,k,\lambda) \), to be the graph with vertex set \( \mathfrak{B} \) in which two vertices are adjacent if and only if their corresponding blocks have exactly two points of \( V \) in common. We define an undirected (resp. directed) hinge to be the multigraph with four vertices which consists of two undirected (resp. directed) 3-cycles which share exactly two vertices in common. An undirected (resp. directed) hinge system of order \( n \) and index \( \lambda \) is a decomposition of \( \lambda K_n \) (resp. \( \lambda{K}_n^* \)) into undirected (resp. directed) hinges. In this paper, we show that each component of the \( B_2 \) block-intersection graph of a simple BIBD\( (n,3,2) \) is 2-edge-connected; this enables us to decompose pure Mendelsohn triple systems and simple 2-fold triple systems into directed and undirected hinge systems, respectively. Furthermore, we obtain a generalisation of the Doyen-Wilson theorem by giving necessary and sufficient conditions for embedding undirected (resp. directed) hinge systems of order \( n \) in undirected (resp. directed) hinge systems of order \( v \). Finally, we determine the spectrum for undirected hinge systems for all indices \( \lambda \geq 2 \) and for directed hinge systems for all indices \( \lambda \geq 1 \).

Vince asked whether for each rational \( r \) between 2 and 4 there was a planar graph of circular chromatic number \( r \). Moser and Zhu showed that the answer is yes, the first for \( 2 < r < 3 \), the second for \( 3 < r < 4 \). This paper gives another family of planar graphs with circular chromatic number between 2 and 3.

We present a new proof that the optimal fast solutions to the gossip problem, for an even number of participants \( n > 2^{\lceil \log_2{n} \rceil} – 2^{\lfloor \lceil \log_2{n} \rceil /2\rfloor} \), require exactly \( \frac{n}{2}\lceil \log_2{n} \rceil \) calls.

We establish that up to an isomorphism there are exactly \(88\) perfect \(1\)-factorizations of \( K_{16} \) having nontrivial automorphism group. We also present some related results.