We extend the basic theory concerning the cycle space of a finite graph to infinite locally finite graphs, using as infinite cycles the homeomorphic images of the unit circle in the graph compactified by its ends. It turns out that the spanning trees allowing such a generalization are precisely the end-faithful spanning trees.

We show that the problem of finding a minimum dominating set in a circle graph is APXhard: there is a constant δ > 0 such that there is no (1 + δ)-approximation algorithm for the minimum dominating set problem on circle graphs unless P = NP. Hence a PTAS for this problem seems unlikely. This hardness result complements the (2 + ε)-approximation algorithm for the problem (Journal of Algorithms, ...

The line graph of an edge-signed graph carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized edge-signed graphs whose line graphs are consistent. Their proof applies Hoede’s relatively difficult characterization of consistent vertex-signed graphs. We give a different, constr...

The fractional and circular chromatic numbers are the two most studied non-integral refinements of number a graph. Starting from definition coloring base graph, which originated in work related to ergodic theory, we formalize notion gyrocoloring graph: vertices colored by translates single Borel set circle group, neighboring receive disjoint translates. corresponding gyrochromatic graph always ...