^{1} Department of Informatics and Mathematical Modeling, Technical University of Denmark^{2} Department of Mathematics, Technical University of Denmark^{3} Discrete mathematics, Department of Mathematics, Technical University of Denmark^{4} King Abdulaziz University^{5} University of Otago^{6} Nanyang Technological University^{7} Department of Applied Mathematics and Computer Science, Technical University of Denmark^{8} King Abdulaziz University

DOI:

10.1016/j.dam.2012.09.009

Abstract:

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provides lower bounds. In this paper, we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightly more than 2q−p cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than 2p+3log2(3p) edges. We also conclude that an Eulerian (even and connected) graph has at most 2q−p cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as 2q−p+p cycles.

Type:

Journal article

Language:

English

Published in:

Discrete Applied Mathematics, 2013, Vol 161, Issue 3, p. 424-429

Keywords:

Minimal codewords; Intersecting codes; Cycle code of graphs