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.

The trace of a degree \( n \) polynomial \( p(x) \) over \( \text{GF}(2) \) is the coefficient of \( x^{n-1} \), and the \emph{subtrace} is the coefficient of \( x^{n-2} \). We derive an explicit formula for the number of irreducible degree \( n \) polynomials over \( \text{GF}(2) \) that have a given trace and subtrace. The trace and subtrace of an element \( \beta \in \text{GF}(2^n) \) are defined to be the coefficients of \( x^{n-1} \) and \( x^{n-2} \), respectively, in the polynomial

\[q(x) = \prod_{i=0}^{n-1} (x + \beta^{2^i}).\]

We also derive an explicit formula for the number of elements of \( \text{GF}(2^n) \) of given trace and subtrace. Moreover, a new two-equation Möbius-type inversion formula is proved.

In this paper, it has been verified, by a computer-based proof, that the smallest size of a complete arc is 12 in \( \text{PG}(2,27) \) and 13 in \( \text{PG}(2,29) \). Also, the spectrum of the sizes of the complete arcs of \( \text{PG}(2,27) \) has been found. The classification of the smallest complete arcs of \( \text{PG}(2,27) \) is given: there are seven non-equivalent 12-arcs, and for each of them, the automorphism group and some geometrical properties are presented. Some examples of complete 13-arcs of \( \text{PG}(2,29) \) are also described.

For a factorization \( F \) of a graph \( G \) into factors \( F_1, F_2, \ldots, F_k \), the chromatic number \( \chi(F) \) of \( F \) is the minimum number of elements \( V_1, V_2, \ldots, V_m \) in a partition of \( V(G) \) such that each subset \( V_i \) \((1 \leq i \leq m)\) is independent in some factor \( F_j \) \((1 \leq j \leq k)\). If \( \chi(F) = m \), then \( F \) is an \( m \)-chromatic factorization.

For integers \( k, m, n \geq 2 \) with \( n \geq m \), the cofactor number \( c_m(k,n) \) is defined as the smallest positive integer \( p \) for which there exists an \( m \)-chromatic factorization \( F \) of the complete graph \( K_p \) into \( k \) factors \( F_1, F_2, \ldots, F_k \) such that \( \chi(F_i) \geq n \) for all integers \( i \) \((1 \leq i \leq k)\). The values of the numbers \( c_m(k,n) \) are investigated for \( m = 3 \) and \( m = 4 \).

The \( k \)-cofactorization number \( \chi_k(G) \) of a graph \( G \) is defined as \( \max\{\chi(F) : F \text{ is a factorization of } G \text{ into } k \text{ factors}\} \). It is shown that \( \chi_k(K_n) \geq \lfloor n^{1/k} \rfloor \) for \( k \geq 2 \) and \( n \geq 4 \). The numbers \( \chi_k(K_n) \) are determined for several values of \( k \) and \( n \).