Maximum odd induced subgraph of a graph concerning its chromatic number
Let f o ( G ) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of G $G$. In 1992, Scott proposed a conjecture that f o ( G ) ≥ n χ ( G ) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph G $G$ of order n $n$ without isolated vertices, where χ ( G ) $\chi (G)$ is the chromatic number of G $G...
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| Published in | Journal of graph theory Vol. 107; no. 3; pp. 578 - 596 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
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Wiley Subscription Services, Inc
01.11.2024
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0364-9024 1097-0118 |
| DOI | 10.1002/jgt.23148 |
Cover
| Abstract | Let
f
o
(
G
) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of
G $G$. In 1992, Scott proposed a conjecture that
f
o
(
G
)
≥
n
χ
(
G
) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph
G $G$ of order
n $n$ without isolated vertices, where
χ
(
G
) $\chi (G)$ is the chromatic number of
G $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that
f
o
(
G
)
≥
2
n
4 ${f}_{o}(G)\ge 2\unicode{x0230A}\frac{n}{4}\unicode{x0230B}$ for a connected graph
G $G$ of order
n $n$. Scott's conjecture is open for graphs with chromatic number at least 3. |
|---|---|
| AbstractList | Let
f
o
(
G
) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of
G $G$. In 1992, Scott proposed a conjecture that
f
o
(
G
)
≥
n
χ
(
G
) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph
G $G$ of order
n $n$ without isolated vertices, where
χ
(
G
) $\chi (G)$ is the chromatic number of
G $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that
f
o
(
G
)
≥
2
n
4 ${f}_{o}(G)\ge 2\unicode{x0230A}\frac{n}{4}\unicode{x0230B}$ for a connected graph
G $G$ of order
n $n$. Scott's conjecture is open for graphs with chromatic number at least 3. Let be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the chromatic number of . In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that for a connected graph of order . Scott's conjecture is open for graphs with chromatic number at least 3. Let fo(G) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of G $G$. In 1992, Scott proposed a conjecture that fo(G)≥nχ(G) ${f}_{o}(G)\ge \frac{n}{\chi (G)}$ for a graph G $G$ of order n $n$ without isolated vertices, where χ(G) $\chi (G)$ is the chromatic number of G $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that fo(G)≥2n4 ${f}_{o}(G)\ge 2\unicode{x0230A}\frac{n}{4}\unicode{x0230B}$ for a connected graph G $G$ of order n $n$. Scott's conjecture is open for graphs with chromatic number at least 3. |
| Author | Wu, Baoyindureng Wang, Tao |
| Author_xml | – sequence: 1 givenname: Tao surname: Wang fullname: Wang, Tao organization: Xinjiang University – sequence: 2 givenname: Baoyindureng orcidid: 0000-0001-7164-3116 surname: Wu fullname: Wu, Baoyindureng email: baoywu@163.com organization: Xinjiang University |
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| Snippet | Let
f
o
(
G
) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of
G $G$. In 1992, Scott proposed a conjecture that
f
o
(
G
)
≥
n
χ
(
G
)... Let be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the... Let fo(G) ${f}_{o}(G)$ be the maximum order of an odd induced subgraph of G $G$. In 1992, Scott proposed a conjecture that fo(G)≥nχ(G) ${f}_{o}(G)\ge... |
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| SubjectTerms | Apexes chromatic number Graph theory Graphs induced subgraphs odd subgraphs |
| Title | Maximum odd induced subgraph of a graph concerning its chromatic number |
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