Edge-maximal graphs without θ7-graphs

M.S.A. Bataineh; M.M.M. Jaradat; I.Y.A. Al-Shboul

doi:10.55937/sut/1315095084

January 2011 Edge-maximal graphs without

{\theta _7}

-graphs

M.S.A. Bataineh, M.M.M. Jaradat, I.Y.A. Al-Shboul

Author Affiliations +

SUT J. Math. 47(1): 91-103 (January 2011). DOI: 10.55937/sut/1315095084

Abstract

Let ${\cal G}(n;{\theta _{2k + 1}}, \ge \delta )$ denote the class of non-bipartite ${\theta _{2k + 1}}$ -free graphs on $n$ vertices and minimum degree at least $\delta$ and let $f(n;{\theta _{2k + 1}}, \ge \delta ) = {\rm{max}}\left\{ {{\cal E}(G):G \in {\cal G}(n;{\theta _{2k + 1}}, \ge \delta )} \right\}$ . In this paper we determine an upper bound of $f(n;{\theta _7}, \ge 25)$ by proving that for large $n$ , $f(n;{\theta _7}, \ge 25) \le {{{{(n - 2)}^2}} \over 4} + 3$ . Our result confirm the conjecture made in [1], ”Some extermal problems in graph theory”, Ph.D thesis, Curtin University of Technology, Australia (2007), in case $k = 3$ and $\delta = 25$ .

ACKNOWLEDGMENT

The authors would like to thank the anonymous referee and editor for their valuable comments and suggestions.

Citation

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M.S.A. Bataineh. M.M.M. Jaradat. I.Y.A. Al-Shboul. "Edge-maximal graphs without ${\theta _7}$ -graphs." SUT J. Math. 47 (1) 91 - 103, January 2011. https://doi.org/10.55937/sut/1315095084