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Barvanja ravninskih in sorodnih družin grafov : doktorska disertacija
Erman, Rok
Predstavimo štiri rezultate v naslednjem vrstnem redu: seznamsko barvanje kvadrata ravninskih grafov, realno označevanje poddružine Kneserjevih grafov, linearno barvanje ravninskih grafov in navadno ...barvanje skoraj ravninskih grafov. Najprej podamo zgornje meje za izbirno število kvadrata ravninskega grafa z maksimalno stopnjo ▫$\Delta \le 4$▫. V posebnem je ▫$G^2$▫ 5-, 6-, 7-, 8-,9-, 12-, 14- ali 15-izbiren, če je ožina grafa ▫$G$▫ vsaj 20, 15, 10, 8, 7, 5, 4 ali 3. Določimo lambda funkcijo ▫$\lambda$▫ za poddružino Kneserjevih grafov. ▫$$\lambda(K(1\ell + 1,2);x, 1) = \begin{cases} 2\ell + (\ell - 1)x & \text{za } x \in \langle 0, 1/\ell),\\ (2\ell^2 + \ell - 1)x & \text{za } x \in \langle 1/\ell, 1),\\ 2\ell^2 + \ell - 1 & \text{za } x \in \langle 1, 3),\\ (2\ell - 2)x + 2\ell^2 - 5\ell + 5 & \text{za } x \ge 3, \end{cases}$$▫ in ▫$$\lambda(K(1\ell + 2,2);x, 1) = \begin{cases} 2\ell + 3\ell x & \text{za } x \in \langle 0, 1/\ell),\\ (2\ell^2 + 3\ell)x & \text{za } x \in \langle 1/\ell, 1),\\ 2\ell^2 + 3\ell & \text{za } x \in \langle 1, 3),\\ (2\ell - 1)x+ 2\ell^2 - 3\ell + 3 & \text{za } x \ge 3. \end{cases}$$▫ Dokažemo, da za linearno kromatično število ▫$\text{lc}(G)$▫ ravninskega grafa ▫$G$▫ z ožino ▫$g \ge 8$▫ in maksimalno stopnjo ▫$\Delta \ge 7$▫ velja neenakost ▫$\text{lc}(G) \le \lceil \frac{\Delta}{2} \rceil +1$▫. Pokažemo tudi, da za vsak graf s prekrižnim številom največ 4 in kličnim številom največ 5 obstaja 5-barvanje. Prav tako dokažemo, da je graf s kličnim številom največ 5, ki mu lahko odstranimo 3 povezave in s tem postane ravninski, nujno 5-obarvljiv.
Type of material - dissertation
; adult, serious
Publication and manufacture - Ljubljana : [R. Erman], 2012
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