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- W4220667456 abstract "Let G $G$ be a simple graph with maximum degree Δ ( G ) ${rm{Delta }}(G)$ and chromatic index χ ′ ( G ) $chi ^{prime} (G)$ . A classical result of Vizing shows that either χ ′ ( G ) = Δ ( G ) $chi ^{prime} (G)={rm{Delta }}(G)$ or χ ′ ( G ) = Δ ( G ) + 1 $chi ^{prime} (G)={rm{Delta }}(G)+1$ . A simple graph G $G$ is called edge- Δ ${rm{Delta }}$ -critical if G $G$ is connected, χ ′ ( G ) = Δ ( G ) + 1 $chi ^{prime} (G)={rm{Delta }}(G)+1$ and χ ′ ( G − e ) = Δ ( G ) $chi ^{prime} (G-e)={rm{Delta }}(G)$ for every e ∈ E ( G ) $ein E(G)$ . Let G $G$ be an n $n$ -vertex edge- Δ ${rm{Delta }}$ -critical graph. Vizing conjectured that α ( G ) $alpha (G)$ , the independence number of G $G$ , is at most n 2 $frac{n}{2}$ . The current best result on this conjecture, shown by Woodall, is α ( G ) < 3 n 5 $alpha (G)lt frac{3n}{5}$ . We show that for any given ε ∈ ( 0 , 1 ) $varepsilon in (0,1)$ , there exist positive constants d 0 ( ε ) ${d}_{0}(varepsilon )$ and D 0 ( ε ) ${D}_{0}(varepsilon )$ such that if G $G$ is an n $n$ -vertex edge- Δ ${rm{Delta }}$ -critical graph with minimum degree at least d 0 ${d}_{0}$ and maximum degree at least D 0 ${D}_{0}$ , then α ( G ) < 1 2 + ε n $alpha (G)lt left(frac{1}{2}+varepsilon right)n$ . In particular, we show that if G $G$ is an n $n$ -vertex edge- Δ ${rm{Delta }}$ -critical graph with minimum degree at least d $d$ and Δ ( G ) ≥ ( d + 1 ) 4.5 d + 11.5 ${rm{Delta }}(G)ge {(d+1)}^{4.5d+11.5}$ , then α ( G ) < 7 n 12 if d = 3 , 4 n 7 if d = 4 , d + 2 + ( d − 1 ) d 3 2 d + 4 + ( d − 1 ) d 3 n < 4 n 7 if d ≥ 19 . $alpha (G)lt left.left{displaystyle begin{array}{cc}frac{7n}{12} & ,text{if},,d=3, frac{4n}{7} & ,text{if},,d=4, frac{d+2+sqrt[3]{(d-1)d}}{2d+4+sqrt[3]{(d-1)d}}nlt frac{4n}{7} & ,text{if},,dge 19.end{array}right.$" @default.
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- W4220667456 date "2022-03-20" @default.
- W4220667456 modified "2023-09-26" @default.
- W4220667456 title "Independence number of edge‐chromatic critical graphs" @default.
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- W4220667456 doi "https://doi.org/10.1002/jgt.22825" @default.
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