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On Petersen's graph theorem

1981
*
Discrete Mathematics
*

The proof is based

doi:10.1016/0012-365x(81)90257-0
fatcat:shtnz3v63beehdms5picafafc4
*on*a useful extension of Tutte's factor*theorem*[4,5], due to JN&Z [3]. For other extensions of*Petersen's**theorem*, see [6,7, $1. ... In thiq paper we prove the following: let G be a*graph*with k edges, wihich js (k -l)-edgeconnectd, and with all valences 3k k. ... From*Theorem*2 we infer ,a corollary*on*regular*graphs*: comlky 1. t G = (V, E) be a (k -l)-edge-connected, k-regular*graph**on*v vertices, a& let 1 G r 6 k be an integer. ...##
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Spanning eularian subgraphs, the splitting Lemma, and Petersen's theorem

1992
*
Discrete Mathematics
*

This result is obtained by applying the Splitting Lemma and

doi:10.1016/0012-365x(92)90587-6
fatcat:bf6caxay7zf6hlr4d7vl6o6psi
*Petersen's**Theorem*.*On*the other hand, it can be viewed as a generalization of this famous*theorem*. ... ., Spanning eulerian subgraphs, the Splitting Lemma, and*Petersen's**Theorem*, Discrete Mathematics 101 (1992) 33-37. ... Let G3 denote this new*graph*. Thus G3 is a connected, bridgeless, cubic*graph*in any case. Applying*Petersen's**Theorem*to G3 we obtain a 2-factor Q c G3. ...##
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Page 860 of American Journal of Mathematics Vol. 69, Issue 4
[page]

1947
*
American Journal of Mathematics
*

*THEOREM*2. The complement of Desarques’

*Graph*is

*Petersen’s*

*Graph*. Proof. ... But

*Petersen’s*

*Graph*has the group given in the

*Theorem*.® *“ The mapping of

*graphs*

*on*surfaces,” Journal of Mathematics and Physics, vol. 16 (1937), pp. 46-75; page 66 and plate I

*on*page 62. ...

##
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Page 493 of Annals of Mathematics Vol. 27, Issue
[page]

1925
*
Annals of Mathematics
*

A PROOF OF

*PETERSEN'S**THEOREM*. 498 and y, being adjacent blue 1-cells, cannot be*on*the same red—blue path. Thus we obtain a contradiction. ... For a*graph*of order two the*theorem*is obvious. This proves the*theorem*.*THEOREM*IV (*Petersen’s**Theorem*). <A regular*graph*of the third degree with fewer than three leaves is colorable. ...##
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Julius Petersen's theory of regular graphs

1992
*
Discrete Mathematics
*

., Julius

doi:10.1016/0012-365x(92)90639-w
fatcat:cbo25io4jrh2zdu2ena37bgs5m
*Petersen's*theory of regular*graphs*, Discrete Mathematics 100 (1992) 157-17s. ... In 1891 the Danish mathematician Julius Petersen (1839-1910) published a paper*on*the factorization of regular*graphs*. ...*Theorem*2. A 4-regular*graph*can be factorized into two 2-factors.*Petersen's*proof reflects the above mentioned geometric view*on**graphs*. ...##
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Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs

1973
*
Journal of combinatorial theory. Series B (Print)
*

For all n ~> 3, regular n-valent nonHamiltonian non-n-edge-colorable

doi:10.1016/s0095-8956(73)80006-1
fatcat:czehutdzpnh35ltg5q6egzbthy
*graphs*with an even ntunber of vertices are constructed. For n @ 5, 6, or 7, these*graphs*are n-connected. ...*Petersen's**graph*has however no 1-factor containing just*one*edge of this type, so we have a contradiction. The result follows by*Theorem*2. ... Each path in*Petersen's**graph*corresponds to either a or b edge-disjoint paths in H,, so for n = 3m, the n-edge-connectedness of H, follows from the fact that*Petersen's**graph*is 3-connected. ...##
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A generalization of Petersen's theorem

1993
*
Discrete Mathematics
*

., A generalization of

doi:10.1016/0012-365x(93)90497-h
fatcat:q2mv2j5oybg4njxxikhoezrv6e
*Petersen's**theorem*, Discrete Mathematics 115 (1993) 277-282.*Petersen's**theorem*asserts that any cubic*graph*with at most 2 cut edges has a perfect matching. ... We generalize this classical result by showing that any cubic*graph*G = (V, E) with at most 1 cut edge has Correspondence to: ...*Petersen's**theorem*and*Theorem*3. ...##
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Page 59 of Annals of Mathematics Vol. 19, Issue
[page]

1917
*
Annals of Mathematics
*

A PROOF OF

*PETERSEN’S**THEOREM*. By H. R. Branana. ...*Petersen’s**theorem*is as follows: A primitive*graph*of degree 3 contains at least three leaves. The*theorem*may be stated in the following form: “cr. ...##
###
Julius Petersen 1839–1910 a biography

1992
*
Discrete Mathematics
*

(iv) The factorization of regular

doi:10.1016/0012-365x(92)90636-t
fatcat:b5dncfdzljfopeifxxrge72q54
*graphs*of odd degree, in particular, the*theorem*that any bridgeless 3-regular*graph*can be decomposed into a l-factor and a 2-factor (*Petersen's**theorem*). ... This paper is the point of departure for*Petersen's**graph*theoretical work. ...##
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A Proof of Petersen's Theorem

1917
*
Annals of Mathematics
*

A PROOF OF

doi:10.2307/1967667
fatcat:weqcptdajrg75bw5shpyhbkz4e
*PETERSEN'S**THEOREM*. BY H. R. BRAHANA. ... This completes the proof of*Petersen's**theorem*. FIG. 2. ...##
###
Page 54 of Mathematical Reviews Vol. , Issue 94a
[page]

1994
*
Mathematical Reviews
*

Several well-known

*theorems**on*maximum matchings, including*Petersen’s*1-factor*theorem*, are generalized.” ... Summary: “*Petersen’s**theorem*asserts that any cubic*graph*with at most 2 cut edges has a perfect matching. ...##
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Regular factors in regular graphs

1993
*
Discrete Mathematics
*

., Regular factors in regular

doi:10.1016/0012-365x(93)90523-v
fatcat:2koej7stvjdetphexpaqmqnpcu
*graphs*, Discrete Mathematics 113 (1993) 269-274. ... Then the*graph*obtained by removing any k -m edges of G, has an m-factor. ... XGS (2) The first results*on*factors in*graphs*were obtained by Petersen [2].*Petersen's*decomposition*theorem*. ...##
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Julius Petersen annotated bibliography

1992
*
Discrete Mathematics
*

Petersens most famous paper, containing the basic theory of

doi:10.1016/0012-365x(92)90637-u
fatcat:3lae6hls4rfgzbdvcauk76yvt4
*graph*factorization, including*Petersen's**Theorem**on*the existence of l-factors in 3-regular*graphs*. ... In the first book*on**graph*theory (Theorie der endlichen und unendlichen Graphen, Leipzig 1936) D. ... Part of*Petersen's*successful set of schooi books, cf. JP 1877a. Tids. ...##
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Parsimonious edge coloring

1996
*
Discrete Mathematics
*

In a

doi:10.1016/0012-365x(94)00254-g
fatcat:ktiirpqqc5frpkp67cbburvpaq
*graph*G of maximum degree A, let y denote the largest fraction of edges that can be A-edge-colored. ... This paper investigates lower bounds for 7 together with infinite families of 13*graphs*in which y is bounded away from 1. ... Acknowledgements We are grateful to Stephen Locke for providing information about prior work*on*this topic. ...##
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Regular factors in vertex-deleted subgraphs of regular graphs

1994
*
Discrete Mathematics
*

The following

doi:10.1016/0012-365x(94)90398-0
fatcat:obnmka5krfctjhaqi3mwifad7i
*theorem*, due to Petersen, is chronologically the first result*on*k-factors in regular*graphs*.*Petersen's**Theorem*[4]. Every 3-regular, 2-connected*graph*has a l-factor. ... There are several following. results which generalize*Petersen's**theorem*.*One*of them is the Biibler's*theorem*[ 11. Every r-regular, (r -1)-edge-connected*graph*of even order has a 1 -factor. ... Let also Hz be the*graph*obtained from a 2r-regular*graph**on*2r + 2 vertices after the deletion of r -1 independent edges. ...
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