preprint Vahan Mkrtchyan author Eckhard Steffen author 15548 An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$ perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a variant of a conjecture of Fan, Raspaud. 2010 eng Mkrtchyan, Vahan, and Eckhard Steffen. “Bricks and Conjectures of Berge, Fulkerson and Seymour.” <i>ArXiv:1003.5782</i>, 2010. V. Mkrtchyan, E. Steffen, ArXiv:1003.5782 (2010). V. Mkrtchyan and E. Steffen, “Bricks and conjectures of Berge, Fulkerson and Seymour,” <i>arXiv:1003.5782</i>. 2010. Mkrtchyan, V., &#38; Steffen, E. (2010). Bricks and conjectures of Berge, Fulkerson and Seymour. <i>ArXiv:1003.5782</i>. @article{Mkrtchyan_Steffen_2010, title={Bricks and conjectures of Berge, Fulkerson and Seymour}, journal={arXiv:1003.5782}, author={Mkrtchyan, Vahan and Steffen, Eckhard}, year={2010} } Mkrtchyan V, Steffen E. Bricks and conjectures of Berge, Fulkerson and Seymour. <i>arXiv:10035782</i>. 2010. Mkrtchyan, Vahan, and Eckhard Steffen. “Bricks and Conjectures of Berge, Fulkerson and Seymour.” <i>ArXiv:1003.5782</i>, 2010. 102022019-06-07T13:10:04Z2022-01-06T06:50:31Z