arXiv subjects

Eckhard Steffen

Publications and source records attributed to Eckhard Steffen.

36 records · Page 2Linked to original sources

arXiv2019

Flows on signed graphs without long barbells

Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this paper, we study whether some basic properties in Tutte's flow theory remain valid for this family of signed graphs. Specifically let $(G,σ)$ be a flow-admissible signed graph without long barbells. We show that it admits a nowhere-zero $6$-flow and that it admits a nowhere-zero modulo $k$-flow if and only if it admits a nowhere-zero integer $k$-flow for each integer $k\geq 3$ and $k \not = 4$. We also show that each nowhere-zero positive integer $k$-flow of $(G,σ)$ can be expressed as the sum of some $2$-flows. For general graphs, we show that every nowhere-zero $\frac{p}{q}$-flow can be normalized in such a way, that each flow value is a multiple of $\frac{1}{2q}$. As a consequence we prove the equality of the integer flow number and the ceiling of the circular flow number for flow-admissible signed graphs without long barbells.

math.CO↗

arXiv2017

Remarks on planar edge-chromatic critical graphs

The only open case of Vizing's conjecture that every planar graph with $Δ\geq 6$ is a class 1 graph is $Δ= 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $Δ= 5$ our result gives insights into the structure of planar $5$-critical graphs, and the result for $Δ=6$ gives support for the truth of Vizing's planar graph conjecture.

math.CO↗

arXiv2017

Petersen cores and the oddness of cubic graphs

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $μ_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. If $G$ is not 3-edge-colorable, then $μ_3(G) \geq 3$. In [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3) (2015) 195-206] it is shown that if $μ_3(G) \not = 0$, then $2 μ_3(G)$ is an upper bound for the girth of $G$. We show that $μ_3(G)$ bounds the oddness $ω(G)$ of $G$ as well. We prove that $ω(G)\leq \frac{2}{3}μ_3(G)$. If $μ_3(G) = \frac{2}{3} μ_3(G)$, then every $μ_3(G)$-core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4-edge-connected cubic graph $G$ with $ω(G) = \frac{2}{3}μ_3(G)$. On the other hand, the difference between $ω(G)$ and $\frac{2}{3}μ_3(G)$ can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer $k\geq 3$, there exists a bridgeless cubic graph $G$ such that $μ_3(G)=k$.

math.CO↗

arXiv2016

Face-degree bounds for planar critical graphs

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist. For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

math.CO↗

arXiv2016

Signed graphs with two negative edges

The presented paper studies the flow number $F(G,σ)$ of flow-admissible signed graphs $(G,σ)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,σ)$ there is a set ${\cal G}(G,σ)$ of cubic graphs such that $F(G, σ) \leq \min \{F(H,σ_H) : (H,σ_H) \in {\cal G}(G)\}$. We prove that $F(G,σ) \leq 6$ if $(G,σ)$ contains a bridge and $F(G,σ) \leq 7$ in general. We prove better bounds, if there is an element $(H,σ_H)$ of ${\cal G}(G,σ)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,σ) \leq 4$ and the bound is tight. If $H$ is 3-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,σ) \leq 6$. Furthermore, if Tutte's 5-Flow Conjecture is true, then $(G,σ)$ admits a nowhere-zero 6-flow endowed with some strong properties.

math.CO↗

arXiv2016

Cores, joins and the Fano-flow conjectures

The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud Conjecture is equivalent to a conjecture proposed in [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78 (2015) 195-206]. Furthermore, we disprove a conjecture proposed in [G. Mazzuoccolo, New conjectures on perfect matchings in cubic graphs, Electron. Notes Discrete Math. 40 (2013) 235-238] and we propose a new version of it under a stronger connectivity assumption. The weak oddness of a cubic graph $G$ is the minimum number of odd components in the complement of a join of $G$. We obtain an upper bound of weak oddness in terms of weak cores, and thus an upper bound of oddness in terms of cores as a by-product.

math.CO↗

arXiv2015

The chromatic spectrum of signed graphs

The chromatic number $χ((G,σ))$ of a signed graph $(G,σ)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= σ(e) c(w)$ for every edge $e = vw$. Let $Σ(G)$ be the set of all signatures of $G$. We study the chromatic spectrum $Σ_χ(G) = \{χ((G,σ))\colon\ σ\in Σ(G)\}$ of $(G,σ)$. Let $M_χ(G) = \max\{χ((G,σ))\colon\ σ\in Σ(G)\}$, and $m_χ(G) = \min\{χ((G,σ))\colon\ σ\in Σ(G)\}$. We show that $Σ_χ(G) = \{k : m_χ(G) \leq k \leq M_χ(G)\}$. We also prove some basic facts for critical graphs. Analogous results are obtained for a notion of vertex-coloring of signed graphs which was introduced by Máčajová, Raspaud, and Škoviera.

math.CO↗

arXiv2015

Circular coloring of signed graphs

Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $χ_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number $χ$. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on $n$ vertices, if the difference is smaller than 1, then there exists $ε_n>0$, such that the difference is at most $1 - ε_n$. We also show that notion of $(k,d)$-colorings is equivalent to $r$-colorings (see (X. Zhu, Recent developments in circular coloring of graphs, in Topics in Discrete Mathematics Algorithms and Combinatorics Volume 26, Springer Berlin Heidelberg (2006) 497-550)).

math.CO↗

arXiv2015

Choosability in signed planar graphs

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in \{3,4,5,6\}$, every signed planar graph without circuits of length $k$ is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.

math.CO↗

arXiv2015

1-factor and cycle covers of cubic graphs

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $μ_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $μ_3(G) \not = 0$, then $2 μ_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with $μ_4(G) = 0$ have a 4-cycle cover of length $\frac{4}{3} |E(G)|$ and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang. We also give a negative answer to a problem of Zhang.

math.CO↗

arXiv2015

Nowhere-zero flows on signed regular graphs

We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{\cal S}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2 + \frac{1}{t} \in {\cal S}(G)$ if and only if $G$ has a $t$-factor. If $G$ has a 1-factor, then $3 \in \overline{\cal S}(G)$, and for every $t \geq 2$, there is a signed $(2t+1)$-regular graph $(H,σ)$ with $ 3 \in \overline{\cal S}(H)$ and $H$ does not have a 1-factor. If $G$ $(\not = K_2^3)$ is a cubic graph which has a 1-factor, then $\{3,4\} \subseteq {\cal S}(G) \cap \overline{\cal S}(G)$. Furthermore, the following four statements are equivalent: (1) $G$ has a 1-factor. (2) $3 \in {\cal S}(G)$. (3) $3 \in \overline{\cal S}(G)$. (4) $4 \in \overline{\cal S}(G)$. There are cubic graphs whose integer flow spectrum does not contain 5 or 6, and we construct an infinite family of bridgeless cubic graphs with integer flow spectrum $\{3,4,6\}$. We show that there are signed graphs where the difference between the integer flow number and the flow number is greater than or equal to 1, disproving a conjecture of Raspaud and Zhu. The paper concludes with a proof of Bouchet's 6-flow conjecture for Kotzig-graphs.

math.CO↗

arXiv2014

Nowhere-zero 5-flows on cubic graphs with oddness 4

Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow.

math.CO↗

arXiv2013

Edge-colorings and circular flow numbers on regular graphs

The paper characterizes $(2t+1)$-regular graphs with circular flow number $2 + \frac{2}{2t-1}$. For $t=1$ this is Tutte's characterization of cubic graphs with flow number 4. The class of cubic graphs is the only class of odd regular graphs where a flow number separates the class 1 graphs from the class 2 graphs. We finally state some conjectures and relate them to existing flow-conjectures.

math.CO↗

arXiv2013

Intersecting 1-factors and nowhere-zero 5-flows

Let $G$ be a bridgeless cubic graph, and $μ_2(G)$ the minimum number $k$ such that two 1-factors of $G$ intersect in $k$ edges. A cyclically $n$-edge-connected cubic graph $G$ has a nowhere-zero 5-flow if (1) $n \geq 6$ and $μ_2(G) \leq 2$ or (2) if $n \geq 5 μ_2(G)-3$

math.CO↗

arXiv2012

Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $ω$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $ω\geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than $ {5/2} ω- 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} ω- 1$, then $G$ has a nowhere-zero 5-flow.

math.CO↗

arXiv2011

Maximum $Δ$-edge-colorable subgraphs of class II graphs

A graph $G$ is class II, if its chromatic index is at least $Δ+1$. Let $H$ be a maximum $Δ$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $Δ$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $Δ\geq3$ can be extended to a maximum $Δ$-edge-colorable subgraph. Simple graphs have a maximum $Δ$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $Δ$-edge-colorable subgraph of a simple graph is always class I.

cs.DM↗

arXiv2010

Measures of edge-uncolorability

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $Δ(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $Δ$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{Δ(G)}{2} \rfloor$, and that this bound is best possible.

cs.DM↗

arXiv2010

Bricks and conjectures of Berge, Fulkerson and Seymour

An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$ perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a variant of a conjecture of Fan, Raspaud.

cs.DM↗