A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matroid analogue of this surface classification. We show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form consisting of the direct sum of the delta-matroids of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Our delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta-matroidal analogues.Keywords: delta-matroid, disc-band surface, handle slide, ribbon graph 2010 MSC: 05B35, 05C10 Overview and backgroundMatroid theory is often thought of as a generalisation of graph theory. W. Tutte famously observed that, "If a theorem about graphs can be expressed in terms of edges and circuits alone it probably exemplifies a more general theorem about matroids" (see [14]). The merit of this point of view is that the more 'tactile' area of graph theory can serve as a guide for matroid theory, in the sense that results and properties for graphs can indicate what results and properties about matroids might hold. In [7] and [8], C. Chun et al. proposed that a similar relationship holds between topological graph theory and delta-matroid theory, writing "If a theorem about embedded graphs can be expressed in terms of its spanning quasi-trees then it probably exemplifies a more general theorem about delta-matroids". Taking advantage of this principle, here we use classical results from surface topology to guide us to a classification of binary delta-matroids.Informally, a ribbon graph is a "topological graph", whose vertices are discs and edges are ribbons, that arises from a regular neighbourhood of a graph in a surface. Formally, a ribbon graph G = (V, E) consists of a set of discs V whose elements are vertices, a set of discs E whose elements are edges, and is such that (i) the vertices and edges intersect in disjoint line segments; (ii) each such line segment lies on the boundary of precisely one vertex and precisely one edge; and (iii) every edge contains exactly two such line segments. We note that ribbon graphs describe exactly cellularly embedded graphs, and refer the reader to [10], or [12] where they are called reduced band

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We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given. Further, we give classes of graphs with the same chromatic and flow polynomials. Mathematics Subject Classification: 05C99

We use a well known concept of proper vertex colouring of a graph to introduce the construction of a chromatic completion graph and its related parameter, the chromatic completion number of a graph. We then give the chromatic completion number of certain classes of cycle derivative graphs and helm graphs. Finally, we discuss further problems for research related to this concept.

We give an alternative method for counting the number of graph compositions of any graph G. In particular we show that counting the number of graph compositions of a graph G is equivalent to counting the number of flats of its cycle matroid. Then we give one condition for non isomorphic graphs to have the same number of graph compositions. (2000): 05C30, 05A18, 05A05, 11B56. Mathematics Subject Classification