Spanning bipartite quadrangulations of even triangulations

Abstract

A triangulation (resp., a quadrangulation) on a surface double-struckF is a map of a loopless graph (possibly with multiple edges) on double-struckF with each face bounded by a closed walk of length 3 (resp., 4). It is easy to see that every triangulation on any surface has a spanning quadrangulation. Kündgen and Thomassen proved that every even triangulation G (ie, each vertex has even degree) on the torus has a spanning nonbipartite quadrangulation, and that if G has sufficiently large edge width, then G also has a bipartite one. In this paper, we prove that an even triangulation G on the torus admits a spanning bipartite quadrangulation if and only if G does not have K 7 as a subgraph, and moreover, we give some other results for the problem.

On perfect matchings in matching covered graphs

“…Lukot'ka and Rollová [7] show that the feasible sets in cubic graphs could be used to show the existence of spanning bipartite qudrangulations (cf. [12]) and certain cycle covers in signed cubic bipartite graphs [7].…”

“…Lukot'ka and Rollová [7] found that the Petersen graph has a nonfeasible edge set that is not switching-equivalent to either ∅ or E G ( ) and believed that an easy characterization of feasible edge sets for regular nonbipartite graphs does not seem possible. More such examples can be found in [12]. But all of these examples are non-3-edge-colorable cubic graphs, which are socalled snarks.…”

Spanning bipartite quadrangulations of triangulations of the projective plane

“…Is there a constant c(Σ) depending only on the surface Σ, such that if the edge-width of a triangulation G of Σ is at least c, then G has a weak 2-coloring? Kündgen and Thomassen [8], Nakamoto, Noguchi and Ozeki [15] independently addressed this question, and considered spanning quadrangulation subgraphs of a given triangulation G of Σ since the following proposition holds (e.g., [15,Proposition 7]). Proposition 2.…”

A characterization of nonfeasible sets in matching covered graphs

“…As far as we know, this concept first appeared in [9]. Lukot'ka and Rollová [9] proved that the feasible sets in cubic graphs could be used to show the existence of spanning bipartite quadrangulations (see [10]) and certain cycle covers in signed cubic bipartite graphs. Clearly, each edge in a matching covered graph forms a trivial feasible set if , while and are two trivial nonfeasible sets.…”

“…They also mentioned in [9] that an easy characterization of nonfeasible sets in regular nonbipartite graphs does not seem to be possible. More such examples can be found in [10]. Since all of these examples are snarks (which belong to the set of cubic graphs of class 2), Lukot'ka and Rollová [9] proposed the following problem.…”