Regular path decompositions of odd regular graphs

Abstract

Kotzig asked in 1979 what are necessary and sufficient conditions for a d-regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1-factor is both necessary and sufficient. Even more, each 1-factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1-factor. We conjecture that existence of a 1-factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1-factors appear to be extendable and we show that for the family of simple 5-regular graphs with no cycles of length 4, all 1-factors are extendable. However, for d>3 we found infinite families of d-regular simple graphs with non-extendable 1-factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5-regular graphs all 1-factors are extendable. ᭧

Decompositions of triangle-free 5-regular graphs into paths of length five

“…Favaron, Genest, and Kouider [16] proved that this condition is not sufficient. For k = 2 (that is, for a 5-regular graph), Favaron, Genest, and Kouider [16] proved that it is sufficient that G contains a perfect matching and no cycles of length four to admit a P 5 -decomposition. Here we prove that every triangle-free 5-regular graph that contains a perfect matching admits a P 5 -decomposition.…”

Codes for distributed storage from 3-regular graphs

“…Table 1 For purposes of comparison, we mention the best known comparable binary linear codes [7] (based on the initial version of Brouwer's tables [12]) meaning codes with the largest minimum distance among all known codes with the same length and dimension. These are [15,6,6], [24,9,8] errors, it does so with access to the entire received word (meaning access to every node) as opposed to using just local information from selected nodes.…”

On path decompositions of 2k-regular graphs

“…A closely related problem to the conjecture of Gallai that has drew great interest [2,7,9,12,13] is as follows: given a family of paths H, is there a decomposition D of G such that each graph in D is isomorphic to a graph in H? We give a step forward towards obtaining constrained path decompositions of even regular graphs.…”