The Z2\mathbb{Z}_2-genus of Kuratowski minors

A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most $2$ common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler $\mathbb{Z}_2$-genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte

Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.Comment: 30 pages, 8 figure

Miscellaneous properties of embeddings of line, total and middle graphs

Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that Full-size image (<1 K) is planar, where Full-size image (<1 K) is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not Full-size image (<1 K) is planar in an Full-size image (<1 K) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs

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