块图的均匀色数的间隙一猜想的一个反证
A disproof of a gap-one conjecture for the equitable chromatic number of block graphs
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中文总结 AI 辅助
该研究针对块图的均匀色数的间隙一猜想，构造出满足特定参数的连通块图，证明均匀色数与$L(G)$的差值无界，从而以强形式否定了该猜想。
中文摘要 AI 辅助
对于图$G$，令$L(G)=\text{max}\{\text{clique number}(G),\text{ceil}((|V(G)|+1)/(\text{minimum independent set size containing vertex }v\text{ over all }v)+1)\}$，其中$\text{clique number}(G)$是团数，$\text{minimum independent set size containing vertex }v\text{ over all }v$是所有顶点$v$中包含$v$的最大独立集的最小规模。Dybizbański、Furmańczyk和Mkrtchyan（《离散应用数学》354卷，2024年，15-28页）猜想每个块图$G$满足$L(G)\text{≤}\text{equitable chromatic number}(G)\text{≤}L(G)+1$，其中$\text{equitable chromatic number}(G)$是$G$的均匀色数。我们以强形式否定该猜想：对每对整数$d\text{≥}2$和$k\text{≥}4d-1$，构造连通块图$G_{d,k}$使得$L(G_{d,k})=k$且$\text{equitable chromatic number}(G_{d,k})=k+d$，因此连通块图上$\text{equitable chromatic number}(G)-L(G)$的差值是无界的。
英文摘要
For a graph $G$, let $L(G)=\max\{ω(G),\lceil (|V(G)|+1)/(α_{\min}(G)+1)\rceil\}$, where $ω(G)$ is the clique number and $α_{\min}(G)$ is the minimum, over all vertices $v$, of the largest size of an independent set containing $v$. Dybizbański, Furmańczyk, and Mkrtchyan (Discrete Appl. Math. 354 (2024), 15--28) conjectured that every block graph $G$ satisfies $L(G)\leqχ_{=}(G)\leq L(G)+1$, where $χ_{=}(G)$ is the equitable chromatic number of $G$. We disprove this conjecture in a strong form. For every pair of integers $d\geq 2$ and $k\geq 4d-1$, we construct a connected block graph $G_{d,k}$ such that $L(G_{d,k})=k$ and $χ_{=}(G_{d,k})=k+d$. Thus the difference $χ_{=}(G)-L(G)$ is unbounded on connected block graphs.
