If \(L\) is a list assignment function and \(\kappa\) is a multiplicity function on the vertices of a graph \(G\), a certain condition on \((G, L, \kappa)\), known as Hall’s multicoloring condition, is obviously necessary for the existence of a multicoloring of the vertices of \(G\). A graph \(G\) is said to be in the class \(MHC\) if it has a multicoloring for any functions \(L\) and \(\kappa\) such that \((G, L, \kappa)\) satisfies Hall’s multicoloring condition. It is known that if \(G\) is in \(MHC\) then each block of \(G\) is a clique and each cutpoint lies in precisely two blocks. We conjecture that the converse is true as well. It is also known that if \(G\) is a graph consisting of two cliques joined at a point then \(G\) is in \(MHC\). We present a new proof of this result which uses common partial systems of distinct representatives, the relationship between matching number and vertex covering number for 3-partite hypergraphs, and Menger’s Theorem.

This paper presents a new approach in the quest for a solution to the \(3x+1\) problem. The method relies on the convergence of the trajectories of the odd positive integers by exploiting the role of the positive integers of the form \(1+4n\), where \(n\) is a non-negative integer.

A cyclic or bicyclic \(9 \times 37\) double Youden rectangle (DYR) is provided for each of the four biplanes with \(k = 9\). These DYRs were obtained by computer search.

For loopless plane multigraphs \(G\), the edge-face chromatic number and the entire chromatic number are asymptotically their fractional counterparts (LP relaxations) as these latter invariants tend to infinity. Proofs of these results are based on analogous theorems for the chromatic index and the total chromatic number, due, respectively, to Kahn [3] and to the first author [6]. Our two results fill in the missing pieces of a complete answer to the natural question: which of the seven invariants associated with colouring the nonempty subsets of \(\{V, E, F\}\) exhibit “asymptotically good” behaviour?

The bin packing problem has been studied extensively since the 1970’s, and it is known to be applicable to many different areas, especially in operations research and computer science. In this paper, we present a variant of the classical bin packing problem, which allows the packing to exceed its bin size but at least a fraction of the last piece is within the bin, and we call it the open-end bin packing problem. This paper is focused on on-line open-end bin packing. An on-line open-end bin packing algorithm is to assign incoming pieces into the bins on-line, that is, there is no information about the sizes of the pieces in future arrivals. An on-line algorithm is optimal if it always produces a solution with the minimum number of bins used for packing. We show that no such optimal algorithm exists. We also present seven efficient on-line algorithms: Next Fit, Random Fit, Worst Fit, Best Fit, Refined Random Fit, Refined Worst Fit, and Refined Best Fit, which give sub-optimal solutions. The performances of these algorithms are studied. A case study for the application of the studied problem is presented, and this is a practical problem on maximizing the savings of using stored-value tickets issued by Kowloon-Canton Railway (KCR), which is one of the major public transportation means in Hong Kong.

We explore the maximum possible toughness among graphs with \(n\) vertices and \(m\) edges in the cases in which \(\lceil \frac{3n}{2}\rceil \leq m < 2n\). In these cases, it is shown that the maximum toughness lies in the interval \([\frac{4}{3}, \frac{3}{2}]\). Moreover, if \(\left\lceil\frac{3n}{2}\right\rceil + 2 \leq m < 2n\), then the value \(\frac{3}{2}\) is achieved. However, if \(m \in \left\{\left\lceil\frac{3n}{2}\right\rceil, \left\lceil\frac{3n}{2}\right\rceil + 1\right\}\), then the maximum toughness can be strictly less than \(\frac{3}{2}\). This provides an infinite family of graphs for which the maximum toughness is not half of the maximum connectivity. The values of maximum toughness are computed for all \(1 \leq n \leq 12\), and some open problems are presented.

A set \(S\) of vertices of a graph \(G = (V, E)\) is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). We define the total domination subdivision number \(sd_{\gamma t}(G)\) to be the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. We give upper bounds on the total domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on \(G\) sufficient to imply that \(sd_{\gamma t}(G) \leq 3\). On the other hand, we show that this constant upper bound does not hold for all graphs. Finally, we show that \(1 \leq sd_{\gamma t}(T) \leq 3\) for any tree \(T\), and characterize the caterpillars \(T\) for which \(sd_{\gamma t}(T) = 3\).

We show that for every \(d \geq 2\), the number of spanning trees of a \(d\)-dimensional grid with \(N\) vertices grows like \(C(d)^N\) for some constant \(C(d)\). Moreover, we show that \(C(d) = 2d-\frac{1}{2}-\frac{5}{16d} + O(d^{-2})\) as \(d\) goes to infinity.

An extended 5-cycle system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a collection of edge-disjoint 5-cycles, 2-tadpoles, and loops that partition the edges of the graph \(K_n^+\) whose vertex set is an \(n\)-set \(V\). In this paper, we show that an extended 5-cycle system of order \(n\) exists for all \(n\) except \(n = 2\) and \(3\).

McMorris, Zaslavsky, and Diny give characterizations of upper bound graphs and double bound graphs in terms of edge clique covers, that is, a family of maximal complete subgraphs that covers all edges. Lundgren and Maybee give a characterization of upper bound graphs using a concept of non-maximal complete subgraphs. In this paper, we present characterizations of double bound graphs and semi-bound graphs in terms of edge covers of non-maximal complete subgraphs.