scholarly journals Edge-disjoint odd cycles in graphs with small chromatic number

1999 ◽  
Vol 49 (3) ◽  
pp. 783-786 ◽  
Author(s):  
Claude Berge ◽  
Bruce Reed
Keyword(s):  
Chromatic Number ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Edge-disjoint odd cycles in 4-edge-connected graphs

2016 ◽  
Vol 119 ◽  
pp. 12-27 ◽  
Author(s):  
Ken-ichi Kawarabayashi ◽  
Yusuke Kobayashi
Keyword(s):  
Connected Graphs ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Edge-disjoint odd cycles in planar graphs

2004 ◽  
Vol 90 (1) ◽  
pp. 107-120 ◽  
Author(s):  
Daniel Král’ ◽  
Heinz-Jürgen Voss
Keyword(s):  
Planar Graphs ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals A Note on Chromatic Number and Induced Odd Cycles

10.37236/5555 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Baogang Xu ◽  
Gexin Yu ◽  
Xiaoya Zha
Keyword(s):  
Chromatic Number ◽  
Bipartite Subgraph ◽  
Odd Cycle ◽  
Odd Cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph $G$ has no odd holes then $\chi(G)\le 2^{2^{\omega(G)+2}}$. Chudnovsky, Robertson, Seymour and Thomas showed that if $G$ has neither $K_4$ nor odd holes then $\chi(G)\le 4$. In this note, we show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then $\chi(G)\le 4$ and $\chi(G)\le 3$ if $G$ has radius at most $3$, and for each vertex $u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions in Plummer and Zha (2014).

scholarly journals A Strengthening on Odd Cycles in Graphs of Given Chromatic Number

2021 ◽  
Vol 35 (4) ◽  
pp. 2317-2327
Author(s):  
Jun Gao ◽  
Qingyi Huo ◽  
Jie Ma
Keyword(s):  
Chromatic Number ◽  
Odd Cycles

scholarly journals Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

2012 ◽  
Vol 21 (6) ◽  
pp. 952-962 ◽  
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Absolute Constant ◽  
The Other ◽  
Asymptotically Optimal ◽  
Edge Disjoint ◽  
Other Hand ◽  
Dense Graphs ◽  
Large Triangle ◽  
Odd Cycles ◽  
Asymptotic Validity

It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

scholarly journals Optimal packings of edge-disjoint odd cycles

2000 ◽  
Vol 211 (1-3) ◽  
pp. 197-202 ◽  
Author(s):  
Claude Berge ◽  
Bruce Reed
Keyword(s):  
Edge Disjoint ◽  
Odd Cycles
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