We describe a concrete data structure, called a sequence-tree, that represents sequences of arbitrary elements, along with associated algorithms that allow single element access and assignment, subsequence extraction (slicing), and concatenation to be done in logarithmic time relative to sequence length. These operations are functional, in the sense that they leave their operand sequences unchanged. For a single sequence, space is linear in the sequence length. Where a set of multiple sequences have been computed by these algorithms, space may be sublinear, because of node sharing. Sequence-trees use immutable, shared, dynamically allocated nodes and thus may require garbage collection, if some of the sequences in a set are abandoned. However, the interconnection of nodes is non-cyclic, so explicitly programmed collection using reference counting is reasonable, should a general-purpose garbage collector be unavailable. Other sequence representations admit only to linear-time algorithms for one or more of the aforementioned operations. Thus sequence-trees give improved performance in applications where all the operations are needed.

This paper is an expository treatment of the Leftover Hash Lemma and some of its applications in cryptography and complexity theory.

We give a necessary and sufficient condition of Hall’s type for a family of sets of even cardinality to be decomposable into two subfamilies having a common system of distinct representatives. An application of this result to partitions of Steiner Triple Systems into small configurations is presented.

In this paper, we construct \(2\)-factorizations of \(K_n\) (\(n\) odd) containing a specified number, \(k\), of \(6\)-cycles, for all integers \(k\) between 0 and the maximum possible expected number of \(6\)-cycles in any \(2\)-factorization, and for all odd \(n\), with no exceptions.

We deal with \((a,d)\)-face antimagic labelings of a certain class of plane quartic graphs. A connected plane graph \(G = (V, E, F)\) is said to be \((a,d)\)-\({face\; antimagic}\) if there exist positive integers \(a\) and \(d\), and a bijection \(g : E(G) \rightarrow \{1,2,…,|E(G)|\}\) such that the induced mapping \(\varphi_g : F(G) \rightarrow {N}\), defined by \(\varphi_g(f) = \sum\{g(e): e \in E(G) \text{ adjacent to face } f\}\), is injective and \(\varphi_g(F) = \{a,a+d,…,a+ (|F(G)| – 1)d\}\).

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\). A vertex labelling \(f : V \rightarrow \{0,1\}\) induces an edge labelling \(\overline{f} : E \rightarrow \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\). In this paper, we show that for every positive integer \(t\) and \(n\) the following families are cordial: (1) Helms \(H_{n}\). (2) Flower graphs \(FL_{n}\). (3) Gear graphs \(G_{n}\). (4) Sunflower graphs \(SFL_{n}\). (5) Closed helms \(CH_{n}\). (6) Generalised closed helms \(CH(t,n)\). (7) Generalised webs \(W(t, n)\).

A cycle \(C\) of a graph \(G\) is called a \(q\)-dominating cycle if every vertex of \(G\) which is not contained in \(C\) is adjacent to at least \(q\) vertices of \(C\). Let \(G\) be a \(k\)-connected graph with \(k \geq 2\). We present a sufficient condition, in terms of the degree sum of \(k + 1\) independent vertices, for \(G\) to have a \(qg\)-dominating cycle. This is an extension of a 1987 result by J.A. Bondy and G. Fan. Furthermore, examples will show that the given condition is best possible.

In an earlier paper [11], we proved that there does not exist any \(\Delta\)-critical graph of even order with five major vertices. In this paper, we prove that if \(G\) is a \(\Delta\)-critical graph of odd order \(2n+1\) with five major vertices, then \(e(G) = n\Delta+1\). This extends an earlier result of Chetwynd and Hilton, and also completes our characterization of graphs with five major vertices. In [9], we shall apply this result to establish some results on class 2 graphs whose core has maximum degree two.

In this paper, uniquely list colorable graphs are studied. A graph \(G\) is said to be uniquely \(k\)-list colorable if it admits a \(k\)-list assignment from which \(G\) has a unique list coloring. The minimum \(k\) for which \(G\) is not uniquely \(k\)-list colorable is called the \(m\)-number of \(G\). We show that every triangle-free uniquely colorable graph with chromatic number \(k+1\) is uniquely \(k\)-list colorable. A bound for the \(m\)-number of graphs is given, and using this bound it is shown that every planar graph has \(m\)-number at most \(4\). Also, we introduce list criticality in graphs and characterize all \(3\)-list critical graphs. It is conjectured that every \(\chi_\ell’\)-critical graph is \(\chi’\)-critical, and the equivalence of this conjecture to the well-known list coloring conjecture is shown.