TY - JOUR

T1 - Bounded-excess flows in cubic graphs

AU - Tarsi, Michael

N1 - Publisher Copyright:
© 2020 Wiley Periodicals, Inc.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - An (r, α)-bounded-excess flow ((r, α)-flow) in an orientation of a graph G = (V, E) is an assignment f : E → [1, r−1], such that for every vertex x ∈ V, (Formula presented.). E+(x), respectively E−(x), is the set of edges directed from, respectively toward x. Bounded-excess flows suggest a generalization of Circular nowhere-zero flows (cnzf), which can be regarded as (r, 0)-flows. We define (r, α) as Stronger or equivalent to (s, β), if the existence of an (r, α)-flow in a cubic graph always implies the existence of an (s, β)-flow in the same graph. We then study the structure of the bounded-excess flow strength poset. Among other results, we define the Trace of a point in the r − α plane by (Formula presented.) and prove that among points with the same trace the stronger is the one with the smaller α (and larger r). For example, if a cubic graph admits a k-nzf (trace k with α = 0), then it admits an (Formula presented.) -flow for every r, 2 ≤ r ≤ k. A significant part of the article is devoted to proving the main result: Every cubic graph admits a (Formula presented.) -flow, and there exists a graph which does not admit any stronger bounded-excess flow. Notice that (Formula presented.) so it can be considered a step in the direction of the 5-flow Conjecture. Our result is the best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture relates only to bridgeless graphs. We also show that if the circular-flow number of a cubic graph is strictly less than 5, then it admits a (Formula presented.) -flow (trace 4). We conjecture such a flow to exist in every cubic graph with a perfect matching, other than the Petersen graph. This conjecture is a stronger version of the Ban-Linial Conjecture [1]. Our work here strongly relies on the notion of Orientable k-weak bisections, a certain type of k-weak bisections. k-Weak bisections are defined and studied by L. Esperet, G. Mazzuoccolo, and M. Tarsi [4].

AB - An (r, α)-bounded-excess flow ((r, α)-flow) in an orientation of a graph G = (V, E) is an assignment f : E → [1, r−1], such that for every vertex x ∈ V, (Formula presented.). E+(x), respectively E−(x), is the set of edges directed from, respectively toward x. Bounded-excess flows suggest a generalization of Circular nowhere-zero flows (cnzf), which can be regarded as (r, 0)-flows. We define (r, α) as Stronger or equivalent to (s, β), if the existence of an (r, α)-flow in a cubic graph always implies the existence of an (s, β)-flow in the same graph. We then study the structure of the bounded-excess flow strength poset. Among other results, we define the Trace of a point in the r − α plane by (Formula presented.) and prove that among points with the same trace the stronger is the one with the smaller α (and larger r). For example, if a cubic graph admits a k-nzf (trace k with α = 0), then it admits an (Formula presented.) -flow for every r, 2 ≤ r ≤ k. A significant part of the article is devoted to proving the main result: Every cubic graph admits a (Formula presented.) -flow, and there exists a graph which does not admit any stronger bounded-excess flow. Notice that (Formula presented.) so it can be considered a step in the direction of the 5-flow Conjecture. Our result is the best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture relates only to bridgeless graphs. We also show that if the circular-flow number of a cubic graph is strictly less than 5, then it admits a (Formula presented.) -flow (trace 4). We conjecture such a flow to exist in every cubic graph with a perfect matching, other than the Petersen graph. This conjecture is a stronger version of the Ban-Linial Conjecture [1]. Our work here strongly relies on the notion of Orientable k-weak bisections, a certain type of k-weak bisections. k-Weak bisections are defined and studied by L. Esperet, G. Mazzuoccolo, and M. Tarsi [4].

KW - Ban-Linial Conjecture

KW - cubic graphs

KW - k-weak bisections

KW - nowhere-zero flows

UR - http://www.scopus.com/inward/record.url?scp=85079052524&partnerID=8YFLogxK

U2 - 10.1002/jgt.22543

DO - 10.1002/jgt.22543

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AN - SCOPUS:85079052524

SN - 0364-9024

VL - 95

SP - 138

EP - 159

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 1

ER -