We introduce a unifying approach for invariants of finite matroids that count mappings to set. The aim this paper is show if the cardinalities with fixed values on restricted set satisfy contraction–deletion rules, then there relation among them can be expressed in terms linear algebra. In way, we study regular chain groups, nowhere-zero flows and tensions graphs, acyclic totally cyclic orienta...

Given a graph $G$ and an odd prime $p$, for mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ ${\mathbb Z}_p$-boundary $b$ of $G$, orientation $\tau$ is called $(f,b;p)$-orientation if the net out $f$-flow same as $b(v)$ in Z}_p$ at each vertex $v\in V(G)$ under $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations closely related to Tutte's nowhere zero 3...

Results related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection %Y of cycles of G which covers all edges of G; U is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of V. By using Seymour’s nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by...

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs: – Erdős–Gallai-type results: characterization of net-degree sequences, – Havel...

We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow. Therefore, a possible minimum counterexample to the 5-flow conjecture has oddness at least 6.

In classical network flow theory, flow being sent from a source to a destination may be split into a large number of chunks traveling on different paths through the network. This effect is undesired or even forbidden in many applications. Kleinberg introduced the unsplittable flow problem where all flow traveling from a source to a destination must be sent on only one path. This is a generaliza...

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant graphs up to vertex switching that contains among its evaluations number proper colorings and nowhere-zero flows. In this, it parallels polynomial graph, which chromatic flow specializations. The tensions (for they are not simply related for graphs) is given in terms trivariate at two distinct points. Interestingl...