Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.
A matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon-Jaeger-Tarsi (AJT) conjecture states that if F is a finite field, with |F| ≥ 4, and A is an element of GL n (F), then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, with |F| ≥ 3, is similar to an AJT matrix. Let AJT n (q) denote the set of n × n, invertible, AJT matrices over a field with q elements. It is shown that the following are equivalent for q ≥ 3: (i) AJT n (q) = GL n (q); (ii) every 2n × n matrix of the form (A|B) t has a nowhere-zero vector in its image, where A, B are n × n, invertible, upper and lower triangular matrices, respectively; and (iii) AJT n (q) forms a semigroup.2000 Mathematics subject classification: primary 15A03; secondary 15A04, 15A23.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.