A mod $(2p+1)$-orientation $D$ is an orientation of $G$ such that $d_D^+(v)-d_D^-(v)\equiv 0 \pmod {2p+1}$ for any vertex $v \in V(G)$. Jaeger conjectured that every $4p$-edge-connected graph has a mod $(2p+1)$-orientation. A graph $G$ is strongly ${\mathbb Z}_{2p+1}$-connected if for every mapping $b: V(G) \mapsto {\mathbb Z}_{2p+1}$ with $\sum_{v\in V(G)}b(v)=0$, there exists an orientation $D$ of $G$ such that $d_D^+(v)-d_D^-(v)= b(v)$ in ${\mathbb Z}_{2p+1}$ for any $v \in V(G)$. A strongly ${\mathbb Z}_{2p+1}$-connected graph admits a mod $(2p+1)$-orientation, and it is a contractible configuration for mod $(2p+1)$-orientation. We prove Jaeger's module orientation conjecture is equivalent to its restriction to bipartite simple graphs and investigate strongly ${\mathbb Z}_{2p+1}$-connectedness of certain bipartite graphs, particularly for $p=2$. We also show that if $G$ is a simple graph with $|V(G)|\ge N(p)= 1152p^4$ and $\min\{\delta(G),\delta(G^c)\}\ge 4p$, then either $G$ or $G^c$ is strongly ${\mathbb Z}_{2p+1}$-connected. When $p=2$, the value of $N(2)$ can be reduced to $N(2) = 80$.

Abstract Let G be a connected graph. A routing in G is a set of fixed paths for all ordered pairs of vertices in G. The forwarding index of G is the minimum of the largest number of paths specified by a routing passing through any vertex of G taken over all routings in G. This article investigates the forwarding index of a wrapped butterfly graph, determines the exact value for the directed case, and gives an upper bound for undirected case. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Abstract Let G be a connected graph of order n. A routing in G is a set of n(n − 1) fixed paths for all ordered pairs of vertices of G. The edge‐forwarding index of G, π(G), is the minimum of the maximum number of paths specified by a routing passing through any edge of G taken over all routings in G, and πΔ,n is the minimum of π(G) taken over all graphs of order n with maximum degree at most Δ. To determine πn−2p−1,n for 4p + 2⌈p/3⌉ + 1 ≤ n ≤ 6p, A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), 249–255]. The set 3 × {1, 2, … , ⌈(4p)/3⌉} can be partitioned into 2p pairs plus singletons such that the set of differences of the pairs is the set 2 × {1, 2, … , p}. This article gives a proof of this conjecture and determines that πn−2p−1,n is equal to 5 if 4p + 2⌈p/3⌉ + 1 ≤ n ≤ 6p and to 8 if 3p + ⌈p/3⌉ + 1 ≤ n ≤ 3p + ⌈(3p)/5⌉ for any p ≥ 2. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 292–296 2004