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### On Petersen's graph theorem

Nathan Linial
1981 Discrete Mathematics
The proof is based on a useful extension of Tutte's factor theorem [4,5], due to JN&Z . 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.  ...

### Spanning eularian subgraphs, the splitting Lemma, and Petersen's theorem

Herbert Fleischner
1992 Discrete Mathematics
This result is obtained by applying the Splitting Lemma and 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.  ...

### 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.  ...

### 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.  ...

### Julius Petersen's theory of regular graphs

Henry Martyn Mulder
1992 Discrete Mathematics
., Julius 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.  ...

### Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs

G.H.J. Meredith
1973 Journal of combinatorial theory. Series B (Print)
For all n ~> 3, regular n-valent nonHamiltonian non-n-edge-colorable 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.  ...

### A generalization of Petersen's theorem

Michel X. Goemans
1993 Discrete Mathematics
., A generalization of 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.  ...

### 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

Jesper Lützen, Gert Sabidussi, Bjarne Toft
1992 Discrete Mathematics
(iv) The factorization of regular 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.  ...

### A Proof of Petersen's Theorem

H. R. Brahana
1917 Annals of Mathematics
A PROOF OF 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.  ...

### Regular factors in regular graphs

P. Katerinis
1993 Discrete Mathematics
., Regular factors in regular 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 . Petersen's decomposition theorem.  ...

### Julius Petersen annotated bibliography

Margit Christiansen, Jesper Lützen, Gert Sabidussi, Bjarne Toft
1992 Discrete Mathematics
Petersens most famous paper, containing the basic theory of 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.  ...

### Parsimonious edge coloring

Michael O. Albertson, Ruth Haas
1996 Discrete Mathematics
In a 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.  ...

### Regular factors in vertex-deleted subgraphs of regular graphs

P. Katerinis
1994 Discrete Mathematics
The following theorem, due to Petersen, is chronologically the first result on k-factors in regular graphs. Petersen's Theorem . 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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