We find new facet-defining inequalities for the linear ordering polytope generalizing the well-known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1 2 }-cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have ...

The matching preclusion number of a graph is the minimum number of neither edges whose deletion in a graph has a neither perfect matching nor an almost perfect matching. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently the conditional matching preclusion number of a graph was introduced to look for sets beyond those induced by a single...

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a speciic innnite sequence of 3-colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that if a (not necessarily maximal) triangle-free graph of order n has minimum degree n=...

A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.

Some properties of homometric sets are known in Music Theory for at least fifty years [16]. We define homometric sets in graphs as follows. Let G = (V,E) denote a simple graph with the vertex set V and the edge set E. Given a vertex set V ′ ⊆ V , the profile of V ′ denotes the multiset of pairwise distances in G between the vertices of V ′. Two disjoint vertex sets of the same size A ⊆ V and B ...

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This “density conjecture” has been proved for the underlying graph Z. We ...

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a c...

A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this "density conjecture" is false when the underlying graph is any of Z2, the complete graph K(n), the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this...

This new topological invariant is applied in the areas of chemical graph theory. In this paper we have obtained Sombor index m−shadow graphs path Dm(Pn), cycle Dm(Cn), ladder Dm(Ln) and tadpole Dm(Tn,p). We also concluded that, m-shadow any regular or finite connected G SO(G)m3

Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q−1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling f...