We show that every bridgeless cubic graph without a Petersen minor has a nowhere-zero 5-flow. This approximates the known 4-flow conjecture of Tutte. A graph has a nowhere-zero k-flow if its edges can be oriented and assigned nonzero elements of the group Zk so that the sum of the incoming values equals the sum of the outcoming ones for every vertex of the graph. An equivalent definition we get...

Central Limit Theorems are known for the Eulerian statistic descent (or excedance) in symmetric group $\SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Theorem over alternating $\AAA_n$ also Carlitz identity using descents. In this paper, we give involving excedances. We extend these to positive elements type B D Coxeter groups. Boroweic Mlotkowski enumerated descents $\DD_n$, similar ...

This is an expository survey of the uses of matrices in the theory of simple graphs with signed edges. A signed simple graph is a graph, without loops or parallel edges, in which every edge has been declared positive or negative. For many purposes the most significant thing about a signed graph is not the actual edge signs, but the sign of each circle (or ‘cycle’ or ’circuit’), which is the pro...

Dominating -color number of a graph is defined as the maximum number of color classes which are dominating sets of and is denoted by d, where the maximum is taken over all -coloring of . In this paper, we discussed the dominating -color number of Generalized Petersen Graphs. We have also discussed the condition under which chromatic number equals dominating -color number of Generalized Pet...

We prove that a connected cographic matroid of a graph G is the bias matroid of a signed graph Σ iff G imbeds in the projective plane. In the case that G is nonplanar, we also show that Σ must be the projective-planar dual signed graph of an actual imbedding of G in the projective plane. As a corollary we get that, if G1, . . . , G29 denote the 29 nonseparable forbidden minors for projective-pl...

The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or −1 to each arc of the n-th order complete graph Kn considered as a digraph. In this paper we show that for n ≥ 3 the base of a primiti...

are considered in this paper. Examples of finite Qn-geometries are constructed for any n i> 2. In particular, we obtain new examples of finite 2-arc-transitive graphs of girth 5, containing Petersen subgraphs (cf. [4], [5] for an enquiry into this class of graphs). Furthermore, we give a complete classification of Q2-geometries using coset enumeration. In particular, we show that they are all f...