In two recent papers [Hafner, Elec. J. Combin. 11 (R77) (2004), 1–33; Ilić, Pace and Magliveras, J. Combin. Math. Combin. Comput. 80 (2012), 267–275] it was shown that the Higman-Sims graph Γ can be decomposed into a disjoint union of five double Petersen graphs. In the second of these papers, it was further shown that all such decompositions fall into a single orbit under the action of sporadi...

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a nonsplit link for every signed embedding. With one ...

There are several graphs H for which the precise structure of graphs that do not contain a minor isomorphic to H is known. In particular, such structure theorems are known for K5 [13], V8 [8] and [7], the cube [3], the octahedron [4], and several others. Such characterizations can often be very useful, e.g., Hadwiger’s conjecture for k = 4 is verified by using the structure for K5-free graphs, ...

By strengthening an edge-decomposition technique for gracefully labelling a generalised Petersen graph, we provide graceful labellings for a new infinite family of such graphs. The method seems flexible enough to provide graceful labellings for many other classes of graphs in the future.

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. use the definition of Ricci curvature on given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation Ollivier, J. Funct. Anal., 2009. A graph Ricci-flat, if it has vanishing edges. show, only Ricci-flat cubic Petersen graph, Triplex and dodecahedral graph. This will correct classification Comm. Anal...

In this paper, we introduce and study a generalized Yosida approximation operator associated to H(·, ·)-co-accretive operator and discuss some of its properties. Using the concept of graph convergence and resolvent operator, we establish the convergence for generalized Yosida approximation operator. Also, we show an equivalence between graph convergence for H(·, ·)-co-accretive operator and gen...

Tutte made the conjecture in 1966 that every 2-connected cubic graph not containing the Petersen graph as a minor is 3-edge-colourable. The conjecture is still open, but we show that it is true in general provided it is true for two special kinds of cubic graphs that are almost planar.