Zero-sum 6-flows in 5-regular graphs

Let $G$ be a graph. A zero-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be a natural number. A zero-sum $k$-flow is a flow with values from the set $\{\pm1, \ldots, \pm(k - 1)\}$. In this paper, we prove that every 5-regular graph admits a zero-sum 6-flow.Comment: arXiv admin note: text overlap with arXiv:1108.2950 by other author

On 1-sum flows in undirected graphs

Let G=(V,E) be a simple undirected graph. For a given set L of the real line, a function omega from E to L is called an L-flow. Given a vector gamma whose coordinates are indexed by V, we say that omega is a gamma-L-flow if for each v in V, the sum of the values on the edges incident to v is gamma(v). If gamma(v)=c, for all v in V, then the gamma-L-flow is called a c-sum L-flow. In this paper we study the existence of gamma-L-flows for various choices of sets L of real numbers, with an emphasis on 1-sum flows. Given a natural k number, a c-sum k-flow is a c-sum flow with values from the set {-1,1,...,1-k, k-1}. Let L be a subset of real numbers containing 0 and let L* be L minus 0 by L*. Answering a question from a recent paper we characterize which bipartite graphs admit a 1-sum R*-flow or a 1-sum Z*-flow. We also show that that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.Comment: 20 pages, 1 figur

Algebraic flow theory of infinite graphs

A problem by Diestel is to extend algebraic flow theory of finite graphs to infinite graphs with ends. In order to pursue this problem, we define an A-flow and non-elusive H-flow for arbitrary graphs and for abelian topological Hausdorff groups H and compact subsets A of H. We use these new definitions to extend several well-known theorems of flows in finite graphs to infinite graphs

Free multiflows in bidirected and skew-symmetric graphs

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ of terminals is called inner Eulerian if each nonterminal node $v$ has even degree (resp. the numbers of edges entering and leaving $v$ are equal). Cherkassky and Lov\'asz showed that the maximum number of pairwise edge-disjoint $T$-paths in an inner Eulerian graph $G$ is equal to $\frac12\sum_{s\in T} \lambda(s)$, where $\lambda(s)$ is the minimum number of edges whose removal disconnects $s$ and $T-\{s\}$. A similar relation for inner Eulerian digraphs was established by Lomonosov. Considering undirected and directed networks with ``inner Eulerian'' edge capacities, Ibaraki, Karzanov, and Nagamochi showed that the problem of finding a maximum integer multiflow (where partial flows connect arbitrary pairs of distinct terminals) is reduced to $O(\log T)$ maximum flow computations and to a number of flow decompositions. In this paper we extend the above max-min relation to inner Eulerian bidirected and skew-symmetric graphs and develop an algorithm of complexity $O(VE\log T\log(2+V^2/E))$ for the corresponding capacitated cases. In particular, this improves the known bound for digraphs. Our algorithm uses a fast procedure for decomposing a flow with O(1) sources and sinks in a digraph into the sum of one-source-one-sink flows.Comment: 21 pages, 4 figures Submitted to a special issue of DA

Koszul Algebras and Flow Lattices

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $\Gamma$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{\Gamma,T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{\Gamma,T}$ modules is isomorphic to the Euclidean lattice $\mathbb Z^{E(\Gamma)}$, and we describe the sublattices of integer cuts and integer flows on $\Gamma$ in terms of the representation theory of $A_{\Gamma,T}$. The grading on $A_{\Gamma,T}$ gives rise to $q$-analogs of the lattices of integer cuts and flows; these $q$-lattices depend non-trivially on the choice of spanning tree. We give a $q$-analog of the matrix-tree theorem, and prove that the $q$-flow lattice of $(\Gamma_1,T_1)$ is isomorphic to the $q$-flow lattice of $(\Gamma_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of $\Gamma_1$ to the edges of $\Gamma_2$ taking the spanning tree $T_1$ to the spanning tree $T_2$. This gives a $q$-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.Comment: 25 pages, minor correction

Nowhere-zero 3-flows in graphs admitting solvable arc-transitive groups of automorphisms

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.Comment: This is the final version to be published in: Ars Mathematica Contemporanea (http://amc-journal.eu/index.php/amc

Odd decompositions of eulerian graphs

We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\equiv |E(G)|\pmod{2}$. We conjecture that a connected $2d$-regular graph of odd order with $d\ge 1$ admits a decomposition into $d$ odd closed trails sharing a common vertex and verify the conjecture for $d\le 3$. The case $d=3$ is crucial for determining the flow number of a signed eulerian graph which is treated in a separate paper (arXiv:1408.1703v2). The proof of our conjecture for $d=3$ is surprisingly difficult and calls for the use of signed graphs as a convenient technical tool.Comment: 15 pages, 3 figure

Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

A unified approach to construct snarks with circular flow number 5

The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5.Comment: 27 pages; submitted for publicatio

The spectrum of the averaging operator on a network (metric graph)

A network is a countable, connected graph X viewed as a one-complex, where each edge [x,y]=[y,x] (x,y in X^0, the vertex set) is a copy of the unit interval within the graph's one-skeleton X^1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X^1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x,y]. There are three natural operators on X : the transition operator P acting on functions on X^0 (the reversible Markov chain associated with the conductances), the averaging operator A over spheres of radius 1 on X^1, and the Laplace operator on X^1 (with Kirchhoff conditions weighted by c(.) at the vertices). The relation between the l^2-spectrum of P and the H^2-spectrum of the Laplacian was described by Cattaneo (Mh. Math. 124, 1997). In this paper we describe the relation between the l^2-spectrum of P and the L^2-spectrum of A