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.

We consider the firefighter problem. We begin by proving that the associated decision problem is NP-complete even when restricted to bipartite graphs. We then investigate algorithms and bounds for trees and square grids.

Face two-colourable triangular embeddings of complete graphs \(K_n\) correspond to biembeddings of Steiner triple systems. Such embeddings exist only if \( n \) is congruent to 1 or 3 modulo 6. In this paper, we present the number of these embeddings for \( n = 13 \).

The resolvable \(2\)-\((14,7,12)\) designs are classified in a computer search: there are 1,363,486 such designs, 1,360,800 of which have trivial full automorphism group. Since every resolvable \(2\)-\((14, 7, 12)\) design is also a resolvable \(3\)-\((14, 7,5)\) design and vice versa, the latter designs are simultaneously classified. The computer search utilizes the fact that these designs are equivalent to certain binary equidistant codes, and the classification is carried out with an orderly algorithm that constructs the designs point by point. As a partial check, a subset of these designs is constructed with an alternative approach by forming the designs one parallel class at a time.