1 Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Department of Mathematics, Technical University of Denmark3 Discrete mathematics, Department of Mathematics, Technical University of Denmark4 King Abdulaziz University5 University of Otago6 Nanyang Technological University7 Department of Applied Mathematics and Computer Science, Technical University of Denmark8 King Abdulaziz University
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.
Discrete Applied Mathematics, 2013, Vol 161, Issue 3, p. 424-429
Minimal codewords; Intersecting codes; Cycle code of graphs