The locally twisted cube $LT{Q}_n$ is a newly introduced interconnection network for parallel computing. As a variant of the hypercube $Q_n$, $LT{Q}_n$ has better properties than $Q_n$ with the same number of links and processors. Yang, Megson and Evans Evans [Locally twisted cubes are $4$-pancyclic, Applied Mathematics Letters, $17(2004),919-925]$ showed that $LT{Q}_n$ contains a cycle of every length from $4$ to $2^n$. In this note, we improve this result by showing that every edge of $LT{Q}_n$ lies on a cycle of every length from $4$ to $2^n$ inclusive.

Necessary and sufficient conditions are given for the existence of a $(K_3+e,\lambda )$-group divisible design of type $g^t{u}^1$.

A $\lambda$-design on $v$ points is a set of $v$ subsets (blocks) of a $v$-set such that any two distinct blocks meet in exactly $\lambda$ points and not all of the blocks have the same size. Ryser’s and Woodall’s $\lambda$-design conjecture states that all $4$-designs can be obtained from symmetric designs by a complementation procedure. In this paper, we establish feasibility criteria for the existence of $\lambda$-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. We use these criteria and a computer to prove that the $\lambda$-design conjecture is true for all $\lambda$-designs with two block sizes with $v\le 90$ and $\lambda \ne 45$.

In this paper, we consider the relationships between the sums of the Fibonacci and Lucas numbers and $1$-factors of bipartite graphs.

We define extended orthogonal sets of $d$-cubes and show that they are equivalent to a class of orthogonal arrays, to geometric nets and a class of codes. As a corollary, an upper bound for the maximal number of $d$-cubes in an orthogonal set is obtained.

For two given graphs $G_1$ and $G_2$, the $Ramseynumber$ $R(G_1,G_2)$ is the smallest integer $n$ such that for any graph $G$ of order $n$, either $G$ contains $G_1$ or the complement of $G$ contains $G_2$. Let $P_n$ denote a path of order $n$ and $W_m$ a wheel of order $m+1$. Chen et al. determined all values of $R(P_n,W_m)$ for $n\ge m-1$. In this paper, we establish the best possible upper bound and determine some exact values for $R(P_n,W_m)$ with $n\le m-2$.

A container $C(x,y)$ is a set of vertex-disjoint paths between vertices $z$ and $y$ in a graph $G$. The width $w(C(x,y))$ and length $L(C(x,y))$ are defined to be $|C(x,y)|$ and the length of the longest path in $C(x,y)$ respectively. The $w$-wide distance $d_w(x,y)$ between $x$ and $y$ is the minimum of $L(C(x,y))$ for all containers $C(x,y)$ with width $w$. The $w$-wide diameter $d_w(G)$ of $G$ is the maximum of $d_w(x,y)$ among all pairs of vertices $x,y$ in $G$, $x\ne y$. In this paper, we investigate some problems on the relations between $d_w(G)$ and diameter $d(G)$ which were raised by D.F. Hsu $[1]$. Some results about graph equation of $d_w(G)$ are proved.

Greedy defining sets have been studied for the first time by the author for graphs. In this paper, we consider greedy defining sets for Latin squares and study the structure of these sets in Latin squares. We give a general bound for greedy defining numbers and linear bounds for greedy defining numbers of some infinite families of Latin squares. Greedy defining sets of circulant Latin squares are also discussed in the paper.

Let $C_n^{(t)}$ denote the cycle with $n$ vertices, and $C_n^{(t)}$ denote the graphs consisting of $t$ copies of $C_n$, with a vertex in common. Koh et al. conjectured that $C_n^{(t)}$ is graceful if and only if $nt\equiv 0,3(\mathrm{mod}4)$. The conjecture has been shown true for $n=3,5,6,7,9,4k$. In this paper, the conjecture is shown to be true for $n=11$.

Let $P(G;\lambda )$ denote the chromatic polynomial of a graph $G$, expressed in the variable $\lambda$. Then $G$ is said to be chromatically unique if $G$ is isomorphic with $H$ for any graph $H$ such that $P(H;\lambda )=P(G;\lambda )$. The graph consisting of $s$ edge-disjoint paths joining two vertices is called an $s$-bridge graph. In this paper, we provide a new family of chromatically unique $5$-bridge graphs.