A 1-tough nonhamiltonian maximal planar graph

AbstractWe construct a maxima' planar graph which is 1-tough but nonhamiltonian. The graph is an answer to Chvátal's question on the existence of such a graph

On the maximum matchings of regular multigraphs

AbstractThe lower bounds on the cardinality of the maximum matchings of regular multigraphs are established in terms of the number of vertices, the degree of vertices and the edge-connectivity of a multigraph. The bounds are attained by infinitely many multigraphs, so are best possible

A Note on the Critical Problem for Matroids

Let M be a matroid representable over GF(q) and S be a subset of its ground set. In this note we prove that S is maximal with the property that the critical exponent c(M|S; q) does not exceed k if and only if S is maximal with the property that c(M · S) ≤ k. In addition, we show that, for regular matroids, the corresponding result holds for the chromatic number. © 1984, Academic Press Inc. (London) Limited. All rights reserved

Size and Energy of Threshold Circuits Computing Mod Functions

Abstract. Let C be a threshold logic circuit computing a Boolean function MODm : {0, 1} n → {0, 1}, where n ≥ 1 and m ≥ 2. Then C outputs "0" if the number of "1"s in an input x ∈ {0, 1} n to C is a multiple of m and, otherwise, C outputs "1." The function MOD2 is the so-called PARITY function, and MODn+1 is the OR function. Let s be the size of the circuit C, that is, C consists of s threshold gates, and let e be the energy complexity of C, that is, at most e gates in C output "1" for any input x ∈ {0, 1} n . In the paper, we prove that a very simple inequality n/(m − 1) ≤ s e holds for every circuit C computing MODm. The inequality implies that there is a tradeoff between the size s and energy complexity e of threshold circuits computing MODm, and yields a lower bound e = Ω((log n − log m)/ log log n) on e if s = O(polylog(n)). We actually obtain a general result on the so-called generalized mod function, from which the result on the ordinary mod function MODm immediately follows. Our results on threshold circuits can be extended to a more general class of circuits, called unate circuits

A Linear Algorithm for Finding Total Colorings of Partial k-Trees

Abstract. A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. However, no efficient algorithm has been known for the total coloring problem on partial k-trees although a polynomial-time algorithm of very high order has been known. In this paper, we give a linear-time algorithm for the total coloring problem on partial k-trees with bounded k

Convex Grid Drawings of Plane Graphs with Rectangular Contours

In a convex drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an n × n grid if G is triconnected or the triconnected component decomposition tree T (G) of G has two or three leaves, where n is the number of vertices in G. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n × n 2 grid if T (G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours