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On the edge-connectivity of C_4-free graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 7، دوره 4، شماره 2، اسفند 2019، صفحه 141-150 اصل مقاله (363.55 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2019.26453.1113 | ||
| نویسنده | ||
| Peter Dankelmann* | ||
| University of Johannesburg | ||
| چکیده | ||
| Let $G$ be a connected graph of order $n$ and minimum degree $\delta(G)$. The edge-connectivity $\lambda(G)$ of $G$ is the minimum number of edges whose removal renders $G$ disconnected. It is well-known that $\lambda(G) \leq \delta(G)$, and if $\lambda(G)=\delta(G)$, then $G$ is said to be maximally edge-connected. A classical result by Chartrand gives the sufficient condition $\delta(G) \geq \frac{n-1}{2}$ for a graph to be maximally edge-connected. We give lower bounds on the edge-connectivity of graphs not containing $4$-cycles that imply that for graphs not containing a $4$-cycle Chartrand's condition can be relaxed to $\delta(G) \geq \sqrt{\frac{n}{2}} +1$, and if the graph also contains no $5$-cycle, or if it has girth at least six, then this condition can be relaxed further, by a factor of approximately $\sqrt{2}$. We construct graphs to show that for an infinite number of values of $n$ both sufficient conditions are best possible apart from a small additive constant. | ||
| کلیدواژهها | ||
| edge-connectivity؛ maximally edge-connected؛ graph | ||
| مراجع | ||
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