• Corpus ID: 248157444

# The chromatic number of (P_5, HVN )-free graphs

@inproceedings{Xu2022TheCN,
title={The chromatic number of (P\_5, HVN )-free graphs},
author={Yian Xu},
year={2022}
}
• Yian Xu
• Published 13 April 2022
• Mathematics
Let G be a graph. We use χ ( G ) and ω ( G ) to denote the chromatic number and clique number of G respectively. A P 5 is a path on 5 vertices, and an HV N is a K 4 together with one more vertex which is adjacent to exactly two vertices of K 4 . In this paper, we show that if G is ( P 5 , HVN )-free, then χ ( G ) ≤ ω ( G ) + 3. This generalizes some results on χ -bounded problem of P 5 -free graphs, and this upper bound is almost sharp as there are many ( P 5 , K 4 )-free graphs with chromatic…
7 Citations

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## References

SHOWING 1-10 OF 37 REFERENCES
Subtrees of a graph and the chromatic number. In The theory and applications of graphs (Kalamazoo
• 1980
On graphs with no induced five‐vertex path or paraglider
• Mathematics
J. Graph Theory
• 2021
The structure of ($P_5, paraglider)-free graphs is studied, and it is shown that every such graph$G$satisfies$\chi(G)\le \lceil \frac{3}{2}\omega(G) \rceil$, where$\chi (G)$and$\omega$are the chromatic number and clique number of$G$, respectively. A tight linear bound to the chromatic number of$(P_5, K_1+(K_1\cup K_3))\$-free graphs
• Mathematics
• 2022
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On the chromatic number of a family of odd hole free graphs
• Mathematics
ArXiv
• 2021
It is proved that for (odd hole, full house)-free graph G, χ(G) ≤ ω(G)+ 1, and the equality holds if and only if ω (G) = 3 and G has H as an induced subgraph.
Coloring of (P5, 4-wheel)-free graphs
• Mathematics
Discret. Math.
• 2022
Perfect coloring and linearly χ-bound P 6 -free graphs
• Mathematics
• 2007
We derive decomposition theorems for P6, K1 + P4-free graphs, P5, K1 + P4-free graphs and P5, K1 + C4-free graphs, and deduce linear χ-binding functions for these classes of graphs (here, Pn (Cn)
Polynomial $$\chi$$χ-Binding Functions and Forbidden Induced Subgraphs: A Survey
• Mathematics
Graphs Comb.
• 2019
This paper addresses perfect graphs, hereditary graphs satisfying the Vizing bound, graphs having linear χ-binding functions and graphs having non-linear polynomial functions, and graph classes defined in terms of forbidden induced subgraphs.