Seifert graphs and the braid index of classical and singular links

We discuss two well known conjectures in knot theory on the relationship between the number of Seifert circles in a link diagram and the writhe of this diagram. One of them, the Malešič and Traczyk conjecture, is formulated in terms of link diagrams and Seifert graphs. By using combinatorial methods, we obtain some new results which confirm partially the Malešič and Traczyk conjecture and disprove its variations formulated in terms of S -graphs. We also indicate the connection of S -graphs with singular link diagrams.