Grothendieck ring class of Banana and Flower graphs

We define a special type of hypersurface varieties inside $\mathbb{P}_k^{n-1}$ arising from connected planar graphs and then find their equivalence classes inside the Gr\"othendieck ring of projective varieties. Then we find a characterization for graphs in order to define irreducible hypersurfaces in general.Comment: To appear in the proceedings of the 6th summer school of topological and algebraic methods in quantum field theory at Villa de Leyva, Colombi

Refining the characterization of projective graphs

Archdeacon showed that the class of graphs embeddable in the projective plane is characterized by a set of 35 excluded minors. Robertson, Seymour and Thomas in an unpublished result found the excluded minors for the class of k-connected graphs embeddable on the projective plane for k = 1,2,3. We give a short proof of that result and then determine the excluded minors for the class of internally 4-connected projective graphs. Hall showed that a 3-connected graph diff_x000B_erent from K5 is planar if and only if it has K3,3 as a minor. We provide two analogous results for projective graphs. For any minor-closed class of graphs C, we say that a set of k-connected graphs E disjoint from C is a k-connected excludable set for C if all but a _x000C_finite number of k-connected graphs not in C have a minor in E. Hall\u27s result is equivalent to saying that {K3,3} is a 3-connected excludable set for the class of planar graphs. We classify all minimal 3-connected excludable sets and fi_x000C_nd one minimal internally 4-connected excludable set for the class of projective graphs. In doing so, we also prove strong splitter theorems for 3-connected and internally 4-connected graphs that could have application to other problems of this type

Disjoint Essential Cycles

AbstractGraphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are structurally characterized

The endomorphism type of certain bipartite graphs and a characterization of projective planes

Fan (in Southeast Asian Bull Math 25, 217-221, 2001) determines the endomorphism type of a finite projective plane. In this note we show that Fan's result actually characterizes the class of projective planes among the finite bipartite graphs of diameter three. In fact, this will follow from a generalization of Fan's theorem and its converse to all finite bipartite graphs with diameter d and girth g such that (1) d + 1 < ga parts per thousand currency sign2d, and (2) every pair of adjacent edges is contained in a circuit of length g