4 research outputs found
Signed circuit -covers of signed -minor-free graphs
Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983):
297-308] proved that every bridgeless ordinary graph has a circuit
-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed
that has a circuit -cover which together implies that has a circuit
-cover for every even integer . The only left case when is
the well-know circuit double cover conjecture. For signed circuit -cover of
signed graphs, it is known that for every integer , there are
infinitely many coverable signed graphs without signed circuit -cover and
there are signed eulerian graphs that admit nowhere-zero -flow but don't
admit a signed circuit -cover. Fan conjectured that every coverable signed
graph has a signed circuit -cover. This conjecture was verified only for
signed eulerian graphs and for signed graphs whose bridgeless-blocks are
eulerian. In this paper, we prove that this conjecture holds for signed
-minor-free graphs. The -cover is best possible for signed
-minor-free graphs