We propose the following conjecture: Let \(m \geq k \geq 2\) be integers such that \(k \mid m\), and let \(T_m\) be a tree on \(m\) edges. Let \(G\) be a graph with \(\delta(G) \geq m+k-1\). Then for every \(Z_k\)-colouring of the edges of \(G\) there is a zero-sum (mod \(k\)) copy of \(T_m\) in \(G\). We prove the conjecture for \(m \geq k = 2\), and explore several relations to the zero-sum Turán numbers.

For any double sequence \((q_{k,n})\) with \(q_{k,0} = 0\), the “summatorial sequence” \((P_{k,n}) = \sum(q_{k,n})\) is defined by \(p_{0,0} = 1\) and \(P_{k,} = \sum_{j=0}^k \sum_{m=1}^n q_{ j,m}P_{k-j,n-m}\) If \(q_{k,n} = 0\) for \(k < n-1\) then there exists a unique sequence \((c_j)\) satisfying the recurrence \(P_{k,n} = \sum_{j=0}^k c_j P_{k-j,k-j,n-m}\) for \(k < n\). We apply this combinatorial recursion to certain counting functions on finite posets. For example, given a set \(A\) of positive integers, let \(P_{k,n}\) denote the number of unlabeled posets with \(n\) points and exactly \(k\) antichains whose cardinality belongs to \(A\), and let \(q_{k,n}\) denote the corresponding number of ordinally indecomposable posets. Then \((P_{k,n})\) is the summatorial sequence of \((q_{k,n})\). If \(2 \in A\) then \((P_{k,n})\) enjoys the above recurrence for \(k < 1\). In particular, for fixed \(k\), there is a polynomial \(p_k\) of degree \(k\) such that \(P_{n,k} = p_k(n)\) for all \(n \geq k\), and \(p_{k,n}\) is asymptotically equal to \(\binom{n-1}{k}\). For some special classes \(A\) and small \(k\), we determine the numbers \(c_k\) and the polynomials \(p_k\) explicitly. Moreover, we show that, at least for small \(k\), the remainder sequences \(p_{k,n} – p_k(n)\) satisfy certain Fibonacci recursions, proving a conjecture of Culberson and Rawlins. Similar results are obtained for labeled posets and for naturally ordered sets.

The paper \([2]\) claimed that a disconnected graph with at least two nonisomorphic components is determined by some three of its vertex deleted subgraphs. While this statement is true, the proof in \([2]\) is incorrect. We give a correct proof of this fact.

It is shown that the necessary conditions for the existence of a \(k\)-cycle system of order \(n\) are sufficient for \(k \in \{20, 24, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48\}\), thus settling the problem for all \(k \leq 50\).

In this paper we study competition graphs of digraphs of restricted degree.
We introduce the notion of restricted competition numbers of graphs.
We complete the characterization of competition graphs of indegree at most 2 and their restricted competition numbers.
We characterize interval \((2,3)\)-graphs and give a recognition algorithm for interval \((2,3)\)-digraphs.
We characterize competition graphs and interval competition graphs of digraphs of outdegree at most \(2\).
The relationship between restricted competition numbers and ordinary competition numbers are studied for several classes of graphs.

A binary linear code of length \(n\), dimension \(k\), and minimum distance at least \(d\) is called an \([n,k,d]\)-code. Let \(d(n,k) = \max \{d : \text{there exists an } [n,k,d]\text{-code}\}\). It is currently known by [6] that \(26 \leq d(66,13) \leq 28\). The nonexistence of a linear \([66,13,28]\)-code is proven.

In this paper, we completely solve the existence problem of \(\text{LOTS}(v)\) (i.e. large set of pairwise disjoint ordered triple systems of order \(v\)).

It is shown that a resolvable BIBD with block size five and index two exists whenever \(v \equiv 5 \pmod{10}\) and \(v \geq 50722395\). This result is based on an updated result on the existence of a BIBD with block size six and index unity, which leaves \(88\) unsolved cases. A construction using difference families to obtain resolvable BIBDs is also presented.

Functions \(c(n)\) and \(h(n)\) which count certain consecutive-integer partitions of a positive integer \(n\) are evaluated, and combinatorial interpretations of partitions with “\(c(n)\) copies of \(n\)” and “\(h(n)\) copies of \(n\)” are given.

J. Leech has posed the following problem: For each integer \(n\), what is the greatest integer \(N\) such that there exists a labelled tree with \(n\) nodes in which the distance between the pairs of nodes include the consecutive values \(1,2,\ldots,N\)? With the help of a computer, we get \(B(n)\) (the number \(N\) for branched trees) for \(2 \leq n \leq 10\) and lower bounds of \(B(11)\) and \(B(12)\). We also get \(U(n)\) (the number \(N\) for unbranched trees) for \(2 \leq n \leq 11\) independently, confirming some results gotten by J. Leech.