recently, e. m'{a}v{c}ajov'{a} and m. v{s}koviera proved that every bidirected eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. this result shows the validity of bouchet's nowhere zero conjecture for eulerian bidirected graphs. in this paper we prove the same theorem in a different terminology and with a short and simple proof. more precisely, we p...

Bouchet conjectured that every bidirected graph which admits a nowhere-zero bidirected flow will admit a nowhere-zero bidirected 6-flow [A. Bouchet, Nowhere-zero integer flows on a bidirected graph, J. Combin. Theory Ser. B 34 (1983) 279–292]. He proved that this conjecture is true with 6 replaced by 216. Zyka proved in his Ph.D dissertation that it is true with 6 replaced by 30. Khelladi prove...

An unoriented flow in a graph, is an assignment of real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph all of whose edges are extraverted. A nowhere-zero unoriented k-flow is an unoriented flow with values from the set {±1, . . . ,±(k − 1)}. It has been conjectured that if a graph has a now...

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...

s Yu Hyunju Isomorphism classes of association schemes induced by Hadamard matrices Every Hadamard matrix H of order n > 1 induces a graph with 4n vertices, called the Hadamard graph Γ(H) of H. Since Γ(H) is a distance-regular graph with diameter 4, it induces a 4-class association scheme (Ω, S) of order 4n. In this article we deal with fission schemes of (Ω, S) under certain conditions, and fo...

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...

Many researchers have devoted themselves to the study of nowhere-zero flows and group connectivity. Recently, Thomassen confirmed the weak 3-flow conjecture, which was further improved by Lovász, Thomassen, Wu and Zhang who proved that every 6-edge-connected graph is Z3-connected. However, Conjectures 1 and 2 are still open. Conjecture 2 implies Conjecture 1 by a result of Kochol that reduces C...