Meno: Robert Luko»ka Real flows in graphs doc. RNDr. Martin ©koviera, PhD. 2006 MMI Real flows, real flow number, flow number, snark, Isaacs snarks A real flow on a graph is a flow with values in $\mathbb{R}$. A real nowhere-zero $r$-flow is a real flow~$\varphi$ with each edge satisfying the condition $1\leq|\varphi (e)|\leq r-1$. The real flow number $\Phi_\mathbb{R}(G)$ of a graph $G$ is the infimum of all reals $r$ such that $G$ has a real nowhere-zero $r$-flow. The purpose of this thesis is threefold. First, we summarize and systematize the fundamental results of real flow theory. We give new proofs of several known results, in particular we present a new direct combinatorial proof of the existence of the minimal real nowhere-zero $r$-flow. Second, we continue in the work of Z. Pan and X. Zhu who showed that for each rational number $r$ between $2$ and $5$ there exist a graph with real flow number $r$ [J. Graph Theory {\bf 49} (2003), 304-318]. We answer their question whether for each rational number \$4

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