New approach to Petersen coloring

doi:10.1016/j.endm.2011.10.026

Electronic Notes in Discrete Mathematics

Volume 38, 1 December 2011, Pages 755-760

https://doi.org/10.1016/j.endm.2011.10.026 Get rights and content

Abstract

Petersen coloring (defined by Jaeger [On graphic-minimal spaces, Ann. Discrete Math. 8 (1980)]) is a mapping from the edges of a cubic graph to the edges of the Petersen graph, so that three edges adjacent at a vertex are mapped to three edges adjacent at a vertex. The existence of such mapping for every cubic bridgeless graph is known to imply the truth of the Cycle double cover conjecture and of the Berge-Fulkerson conjecture.

We develop Jaegerʼs alternate formulation of Petersen coloring in terms of special five-edge colorings. We suggest a weaker conjecture, and provide new techniques to solve it.

On a related note, we provide a counterexample to a stronger conjecture by DeVos, Nešetřil, and Raspaud [On edge-maps whose inverse preserves flows and tensions, Graph Theory in Paris, 2006] that asked for an oriented version of Petersen coloring.

References (7)

F. Jaeger
On graphic-minimal spaces
Ann. Discrete Math.
(1980)
D. Kráľ et al.
Projective, affine, and abelian colorings of cubic graphs
European J. Combin.
(2009)
E. Máčajová et al.
Fano colourings of cubic graphs and the Fulkerson conjecture
Theoret. Comput. Sci.
(2005)

There are more references available in the full text version of this article.

Cited by (20)

Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph
2024, Applied Mathematics and Computation
In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is normal if every edge of G is either rich or poor. In this paper we consider snarks $\tilde{G}$ obtained by a simple superposition of edges and vertices of a cycle C in a snark G. For an even cycle C we show that a normal coloring of G can be extended to a normal coloring of $\tilde{G}$ without changing colors of edges outside C in G. An interesting remark is that this is in general impossible for odd cycles, since the normal coloring of a Petersen graph $P_{10}$ cannot be extended to a superposition of $P_{10}$ on a 5-cycle without changing colors outside the 5-cycle. On the other hand, as our colorings of the superpositioned snarks introduce 18 or more poor edges, we are inclined to believe that every bridgeless cubic graph distinct from $P_{10}$ has a normal coloring with at least one poor edge and possibly with at least 6 if we also exclude the Petersen graph with one vertex being truncated.
H-colorings for 4-regular graphs
2024, Discrete Mathematics
For a vertex a in H, we denote by $\partial_{H} (a)$ the set of edges incident with a. An H-coloring of a graph G is a proper edge-coloring $f : E (G) \to E (H)$ such that for each vertex u in G, there exists a vertex a in H such that $f (\partial_{G} (u)) = \partial_{H} (a)$ . In this paper, we introduce some graphs H such that H-colorability of 4-regular graphs is meaningful and related to some of their properties, such as an even cycle decomposition of size 3 and an even 2-factor. In addition, we formulate a conjecture on H-coloring of the line graph of some cubic graphs. Partial solutions to the conjecture are also presented.
Partially normal 5-edge-colorings of cubic graphs
2021, European Journal of Combinatorics
Citation Excerpt :
A cubic graph has a Petersen-coloring if and only if it has a normal 5-edge-coloring. Considering that a normal 5-edge-coloring requires each edge to be normal, Šámal [18] presented a weaker problem approximate to the Petersen coloring conjecture, that is, to search for a proper 5-edge-coloring such that the normal edges are as many as possible. Here, such a coloring is called a partially normal 5-edge-coloring.
In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the five edges incident with an end of $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $μ_{3}$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $| E (G) | - μ_{3} (G)$ (which is no less than $\frac{27}{35} | E (G) |$ ) edges are normal. This result improves on some earlier results of Bílková and Šámal.
Normal 6-edge-colorings of some bridgeless cubic graphs
2020, Discrete Applied Mathematics
In an edge-coloring (proper) of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly three or exactly five distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal $k$ -edge-coloring of a cubic graph is an edge-coloring with $k$ colors such that each edge of the graph is normal. We denote by $χ_{N}^{'} (G)$ the smallest $k$ , for which $G$ admits a normal $k$ -edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving $χ_{N}^{'} (G) \leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge–Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal 7-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal 6-edge-coloring. Finally, we show that any bridgeless cubic graph $G$ admits a 6-edge-coloring such that at least $\frac{7}{9} \cdot | E |$ edges of $G$ are normal.
Normal 5-edge coloring of some more snarks superpositioned by the Petersen graph
2023, arXiv
Normal 5-edge-coloring of some snarks superpositioned by Flower snarks
2023, arXiv

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¹: Partially supported by grant GA ČR P201/10/P337. Institute for Theoretical Computer Science is supported as project 1M0545 by Ministry of Education of the Czech Republic.

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New approach to Petersen coloring

Abstract

Ann. Discrete Math.

European J. Combin.

Theoret. Comput. Sci.