Hamiltonicity of edge-chromatic critical graphs

Cao, Yan; Chen, Guantao; Jiang, Suyun; Liu, Huiqing; Lu, Fuliang

doi:10.1016/j.disc.2020.111881

Cited by 2 publications

(2 citation statements)

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“…Woodall [21,22] treated color φ yz ( ) of the edge yz as a missing color in φ y ( ) if d z ( ) is "small." This technique was used in [1][2][3][4] in their work on Vizing's average degree conjecture and the hamiltonian property of Δ-critical graphs. For a vertex…”

Section: Adding Colors Of Edges Incident With Vertices With Small Deg...mentioning

confidence: 99%

“…Let

G

be a

{\rm{\Delta }}

‐critical graph,

e=xy\in E(G)

and

\varphi \in {{\mathscr{C}}}^{{\rm{\Delta }}}(G-e)

. Woodall [21, 22] treated color

\varphi (yz)

of the edge

yz

as a missing color in

\bar{\varphi }(y)

d(z)

is “small.” This technique was used in [1–4] in their work on Vizing's average degree conjecture and the hamiltonian property of

{\rm{\Delta }}

‐critical graphs. For a vertex

v\in V(G)

, let

\begin{matrix} false0.33em \\ right φ_{x}^{s} MathClass-open(v MathClass-close) & center = & left {φ (v w) : w \neq x and d (w) \leq true \frac{1}{2} (normalΔ (G \end{matrix}

…”

Section: Adding Colors Of Edges Incident With Vertices With Small Deg...mentioning

confidence: 99%

See 1 more Smart Citation

Double Vizing fans in critical class two graphs

Cao

Chen

2022

Journal of Graph Theory

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Let G $G$ be a simple graph and χ′(G) $\chi ^{\prime} (G)$ be the chromatic index of G $G$. We call G $G$ a normalΔ ${\rm{\Delta }}$‐critical graph if χ′(G−e)=χ′(G)−1=normalΔ $\chi ^{\prime} (G-e)=\chi ^{\prime} (G)-1={\rm{\Delta }}$ for every edge e $e$ of G $G$, where normalΔ ${\rm{\Delta }}$ is maximum degree of G $G$. Let e=xy $e=xy$ be an edge of normalΔ ${\rm{\Delta }}$‐critical graph G $G$ and φ $\varphi $ be an (proper) edge normalΔ ${\rm{\Delta }}$‐coloring of G−e $G-e$. An e‐fan is a sequence Fe=(x,e,y,e1,z1,…,ep,zp) ${F}^{e}=(x,e,y,{e}_{1},{z}_{1},\ldots ,{e}_{p},{z}_{p})$ of alternating vertices and distinct edges such that edge ei ${e}_{i}$ is incident with x $x$ or y $y$, zi ${z}_{i}$ is another endvertex of ei ${e}_{i}$ and φ(ei) $\varphi ({e}_{i})$ is missing at a vertex before zi ${z}_{i}$ for each i $i$ with 1≤i≤p $1\le i\le p$. In this paper, we prove that if min{d(x),d(y)}≤normalΔ−1 $\min \{d(x),d(y)\}\le {\rm{\Delta }}-1$, where d(x) $d(x)$ and d(y) $d(y)$ denote the degrees of vertices x $x$ and y $y$, respectively, then colors missing at different vertices of V(Fe) $V({F}^{e})$ are distinct. Clearly, a Vizing fan is an e $e$‐fan with the restricting that all edges ei ${e}_{i}$ being incident with one fixed endvertex of edge e $e$. This result gives a common generalization of several recently developed new results on multifan, double fan, Kierstead path of four vertices, and broom. By treating some colors of edges incident with vertices of low degrees as missing colors, Kostochka and Stiebitz introduced C $C$‐fan. In this paper, we also generalize the C $C$‐fan from centered at one vertex to one edge.

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