scholarly journals Flows in Signed Graphs with Two Negative Edges

10.37236/4458 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Edita Rollová ◽  
Michael Schubert ◽  
Eckhard Steffen
Keyword(s):  
Signed Graph ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Signed Graphs ◽  
Flow Number ◽  
Edge Connectivity ◽  
Cyclic Edge Connectivity

The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set of cubic graphs obtained from $(G,\sigma)$ such that the flow number of $(G,\sigma)$ does not exceed the flow number of any of the cubic graphs. We prove that $F(G,\sigma) \leq 6$ if $(G,\sigma)$ contains a bridge, and $F(G,\sigma) \leq 7$ in general. We prove better bounds, if there is a cubic graph $(H,\sigma_H)$ obtained from $(G,\sigma)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,\sigma) \leq 4$ and the bound is tight. If $H$ is $3$-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,\sigma) \leq 6$. Furthermore, if Tutte's $5$-Flow Conjecture is true, then $(G,\sigma)$ admits a nowhere-zero $6$-flow endowed with some strong properties.

An Algorithm for Cyclic Edge Connectivity of Cubic Graphs

2004 ◽  
pp. 236-247 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Jan Kára ◽  
Daniel Král’ ◽  
Ondřej Pangrác
Keyword(s):  
Cubic Graphs ◽  
Edge Connectivity ◽  
Cyclic Edge Connectivity

Uniform cyclic edge connectivity in cubic graphs

COMBINATORICA ◽  
1991 ◽  
Vol 11 (2) ◽  
pp. 81-96 ◽  
Author(s):  
R. E. L. Aldred ◽  
D. A. Holton ◽  
Bill Jackson
Keyword(s):  
Cubic Graphs ◽  
Edge Connectivity ◽  
Cyclic Edge Connectivity

scholarly journals Determining the Circular Flow Number of a Cubic Graph

10.37236/9607 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Robert Lukoťka
Keyword(s):  
Polynomial Algorithms ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Flow Number ◽  
Circular Flow ◽  
Bridgeless Graph

A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number, $\phi_c(G)$, of $G$ is the infimum over all values $r$ such that $G$ admits a nowhere-zero $r$-flow. A flow has its underlying orientation. If we subtract the number of incoming and the number of outgoing edges for each vertex, we get a mapping $V(G) \to \mathbb{Z}$, which is its underlying balanced valuation. In this paper we describe efficient and practical polynomial algorithms to turn balanced valuations and orientations into circular nowhere zero $r$-flows they underlie with minimal $r$. Using this algorithm one can determine the circular flow number of a graph by enumerating balanced valuations. For cubic graphs we present an algorithm that determines $\phi_c(G)$ in case that $\phi_c(G) \leqslant 5$ in time $O(2^{0.6\cdot|V(G)|})$. If $\phi_c(G) > 5$, then the algorithm determines that $\phi_c(G) > 5$ and thus the graph is a counterexample to Tutte's $5$-flow conjecture. The key part is a procedure that generates all (not necessarily proper) $2$-vertex-colourings without a monochromatic path on three vertices in $O(2^{0.6\cdot|V(G)|})$ time. We also prove that there is at most $2^{0.6\cdot|V(G)|}$ of them.

Average Edge-Connectivity of Cubic Graphs

2021 ◽  
pp. 2142002
Author(s):  
Miaomiao Zhuo ◽  
Qinqin Li ◽  
Baoyindureng Wu ◽  
Xinhui An
Keyword(s):  
Disjoint Paths ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Edge Connectivity ◽  
Edge Disjoint

In this paper, we consider the concept of the average edge-connectivity [Formula: see text] of a graph [Formula: see text], defined to be the average, over all pairs of vertices, of the maximum number of edge-disjoint paths connecting these vertices. Kim and O previously proved that [Formula: see text] for any connected cubic graph on [Formula: see text] vertices. We refine their result by showing that [Formula: see text] We also characterize the graphs where equality holds.

scholarly journals Spectral Threshold for Extremal Cyclic Edge-Connectivity

2021 ◽  
Author(s):  
Sinan G. Aksoy ◽  
Mark Kempton ◽  
Stephen J. Young
Keyword(s):  
Edge Connectivity ◽  
Cyclic Edge Connectivity

Perfect Matching Index versus Circular Flow Number of a Cubic Graph

2021 ◽  
Vol 35 (2) ◽  
pp. 1287-1297
Author(s):  
Edita Máčajová ◽  
Martin Škoviera
Keyword(s):  
Perfect Matching ◽  
Cubic Graph ◽  
Flow Number ◽  
Circular Flow ◽  
Matching Index

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

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