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Barvanja ravninskih in sorodnih družin grafov : doktorska disertacijaErman, RokPredstavimo 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, seriousPublication and manufacture - Ljubljana : [R. Erman], 2012Language - slovenianCOBISS.SI-ID - 263847680
Author
Erman, Rok
Other authors
Škrekovski, Riste
Topics
Grafi |
Barvanje |
Disertacije |
teorija grafov |
izberljivost |
kvadrat grafa |
ožina |
maksimalna stopnja |
kromatično število |
prekrižno število |
klično število |
linerano barvanje
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Erman, Rok | 29585 |
Škrekovski, Riste | 15518 |
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