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 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 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

Packing Chromatic Number of Cycle Related Graphs

2014 ◽  
Vol 4 (1) ◽  
pp. 27
Author(s):  
Albert William ◽  
Roy Santiago ◽  
Indra Rajasingh
Keyword(s):  
Chromatic Number ◽  
Packing Chromatic Number

Packing coloring on subdivision-vertex and subdivision-edge join of cycle Cm with path Pn

2020 ◽  
pp. 627-634
Author(s):  
K. Rajalakshmi ◽  
M. Venkatachalam ◽  
M. Barani ◽  
D. Dafik
Keyword(s):  
Chromatic Number ◽  
Path Graph ◽  
Packing Chromatic Number ◽  
Cycle Graph

The packing chromatic number $\chi_\rho$ of a graph $G$ is the smallest integer $k$ for which there exists a mapping $\pi$ from $V(G)$ to $\{1,2,...,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. In this paper, the authors find the packing chromatic number of subdivision vertex join of cycle graph with path graph and subdivision edge join of cycle graph with path graph.

scholarly journals On metric chromatic number of comb product of ladder graph

2021 ◽  
Vol 1836 (1) ◽  
pp. 012026
Author(s):  
M Y Rohmatulloh ◽  
Slamin ◽  
A I Kristiana ◽  
Dafik ◽  
R Alfarisi
Keyword(s):  
Chromatic Number ◽  
Ladder Graph
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