![](/files/happy5.png)
partition
Dividing up the unrestricted partitions ★★
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
Keywords: congruence properties; partition
Friendly partitions ★★
Author(s): DeVos
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
Unfriendly partitions ★★★
If is a graph, we say that a partition of
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.
Keywords: coloring; infinite graph; partition
Aharoni-Berger conjecture ★★★
![$ M_1,\ldots,M_k $](/files/tex/368dea3f4a89576f8e4eebf3241a6ef062e5b5d9.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $](/files/tex/d896134dc1e4119543db8e0baaebe50b9bb34085.png)
![$ \{X_1,\ldots,X_k\} $](/files/tex/af99ea0d6ceb5907ffb549d85a7c7e711c6b91c7.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ X \subseteq E $](/files/tex/3b78a133c1f7e4ab67c2c7c5a7fbba992d13e9dc.png)
![$ |X| = \ell $](/files/tex/6ed3addf64c5678ff6cd1add3575f29c9a04af48.png)
![$ M_i $](/files/tex/71c6239abfbaf80e551a379622380b245aaae23a.png)
Keywords: independent set; matroid; partition
Bounded colorings for planar graphs ★★
Author(s): Alon; Ding; Oporowski; Vertigan
![$ f : {\mathbb N} \rightarrow {\mathbb N} $](/files/tex/e5839c90f2b5ca6fe2f58de668c9549b3ad831bd.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ \{V_1,V_2,V_3\} $](/files/tex/4743659cb13aa2fd42df6291dc9839397a771197.png)
![$ i=1,2,3 $](/files/tex/a551cf18cf10c10840b4155cdb12c330c8fec96b.png)
![$ V_i $](/files/tex/af854be1f03aac481e0a165c3908976d4b5b0aa0.png)
![$ f(d) $](/files/tex/eb1c96d175a846e74b707abbc2eabf3ea4a2d7b2.png)
Keywords: coloring; partition; planar graph
Linial-Berge path partition duality ★★★
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords: coloring; directed path; partition
![Syndicate content Syndicate content](/misc/feed.png)