20.500.12556/DKUM-51432 Simpler multicoloring of triangle-free hexagonal graphs Enostavnejše večbarvanje heksagonalnih grafov brez trikotnikov Preslikavo ▫$f colon V(G)to 2^{{1,.,n}}$▫, za katero velja ▫$|f(v)| ge p(v)$▫ za vsako točko ▫$v in V(G)$▫ in ▫$f(v) cap f(u) = emptyset$▫ za poljubni sosedi ▫$u$▫ in ▫$v$▫ grafa ▫$G$▫, imenujemo dobro ▫$n-[p]$▫barvanje grafa ▫$G$▫. Najmanjše naravno število, za katero obstaja dobro ▫$n-[p]$▫barvanje grafa ▫$G$▫, ▫$chi_p(G)$▫, imenujemo uteženo kromatično število grafa ▫$G$▫. Iskanje uteženega kromatičnega števila za inducirane podgrafe trikotniške mreže (imenovane heksagonalni grafi) ima aplikacije v celičnih mrežah. Uteženo kromatično število grafa ▫$G$▫, ▫$omega_p(G)$▫, je enako maksimalni uteži klike grafa ▫$G$▫, kjer utež klike predstavlja vsoto uteži njenih točk. McDiarmid in Reed (2000) sta postavila domnevo, da za poljuben heksagonalen graf brez trikotnikov velja ▫$chi_p(G) le (9/8)omega_p(G) + C$▫. V članku je podan algoritem, ki poda dobro ▫$7-[3]$▫barvanje poljubnega heksagonalnega grafa brez trikotnikov, ki aplicira neenakost ▫$chi_p(G) le (7/6)omega_p(G) + C$▫. Naš rezultat podaja krajšo alternativo induktivnega dokaza Haveta (2001) in izboljša kratek dokaz Sudepa in Vishwanathana (2005), ki sta dokazala obstoj ▫$14-[6]$▫barvanja. (Omeniti je potrebno, da v sklopu našega dokaza uporabimo izrek o štirih barvah.) Vsi koraki algoritma so linearni glede na ▫$|V(G)|$▫, razen 4-barvanje ravninskega grafa. Novi pristop lahko v prihodnje pripomore k dokazovanju domneve McDiarmida in Reeda (2000). Given a graph ▫$G$▫ and a demand function ▫$p colon V(G)to mathbb{N}$▫, a proper ▫$n-[p]$▫coloring is a mapping ▫$f colon V(G)to 2^{{1,.,n}}$▫ such that ▫$|f(v)| ge p(v)$▫ for every vertex ▫$v in V(G)$▫ and ▫$f(v) cap f(u) = emptyset$▫ for any two adjacent vertices ▫$u$▫ and ▫$v$▫. The least integer ▫$n$▫ for which a proper ▫$n-[p]$▫coloring exists, ▫$chi_p(G)$▫, is called multichromatic number of ▫$G$▫. Finding multichromatic number of induced subgraphs of the triangular lattice (called hexagonal graphs) has applications in cellular networks. Weighted clique number of a graph ▫$G$▫, ▫$omega_p(G)$▫, is the maximum weight of a clique in ▫$G$▫, where the weight of a clique is the total demand of its vertices. McDiarmid and Reed (2000) conjectured that ▫$chi_p(G) le (9/8)omega_p(G)+C$▫ for triangle-free hexagonal graphs, where ▫$C$▫ is some absolute constant. In this article, we provide an algorithm to find a ▫$7-[3]$▫coloring of triangle-free hexagonal graphs (that is, when ▫$p(v) = 3$▫ for all ▫$v in V(G)$▫), which implies that ▫$chi_p(G) le (7/6)omega_p(G) + C$▫. Our result constitutes a shorter alternative to the inductive proof of Havet (2001) and improves the short proof of Sudeep and Vishwanathan (2005), who proved the existence of a ▫$14-[6]$▫coloring. (It has to be noted, however, that our proof makes use of the 4-color theorem.) All steps of our algorithm take time linear in ▫$|V(G)|$▫, except for the 4-coloring of an auxiliary planar graph. The new techniques may shed some light on the conjecture of McDiarmid and Reed (2000). matematika teorija grafov aproksimacijski algoritem barvanje grafov dodeljevanje frekvenc celične mreže mathematics graph algorithm graph theory approximation algorithm graph coloring frequency planning cellular networks true false false Angleški jezik Angleški jezik Delo ni kategorizirano 2015-07-10 14:46:29 2015-07-10 14:46:29 2022-05-30 03:05:48 2019-07-11 07:18:50 2012 0 0 Str. 181-187 iss. 1 Vol. 312 2012 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 16067161 6917907 0012-365X URN:SI:UM:DK:IOFJJWSU http://dx.doi.org/10.1016/j.disc.2011.07.031 1 https://dk.um.si/Dokument.php?lang=slv&id=75992 Fakulteta za organizacijske vede 0 0 0