# On specific factors in graphs

## TL;DR

This paper explores the existence of spanning forests with prescribed parity degree conditions on vertices in multigraphs, extending the concepts of even and odd factors in a unified framework.

## Contribution

It generalizes the concepts of even-factors and odd-factors, providing a unified approach to degree parity constraints in spanning forests of multigraphs.

## Key findings

- Established conditions for spanning forests with parity degree constraints.
- Unified the concepts of even-factors and odd-factors.
- Extended known results to more complex graph structures.

## Abstract

It is well known that if $G = (V, E)$} is a multigraph and $X\subset V$ is a subset of even order, then $G$ contains a spanning forest $H$ such that each vertex from $X$ has an odd degree in $H$ and all the other vertices have an even degree in $H$. This spanning forest may have isolated vertices. If this is not allowed in $H$, then the situation is much more complicated. In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form.