CIRCUIT COVERS OF SIGNED EULERIAN GRAPHS

We continue the study of circuit covers of signed graphs initiated by Maeajova et al. [J. Graph Theory, 81 (2016), pp. 120-133] by investigating signed circuit covers of signed Eulerian graphs. A signed circuit cover of a signed graph is a collection of signed circuits such that each edge of the signed graph belongs to at least one of them. We prove that every signed Eulerian graph G that admits a signed circuit cover has one of length at most 3/2 . vertical bar E(G)vertical bar. We show that the bound is tight and characterize those graphs that reach it. This result stands in a sharp contrast with the unsigned case where the bound is 1 . vertical bar E(G)vertical bar (by the classical result of Veblen). For signed Eulerian graphs with an even number of negative edges we establish a better bound of 4/3 . vertical bar E(G)vertical bar and show that it is also tight.