1. The independence coloring game on graphsBoštjan Brešar, Daša Mesarič Štesl, 2022, izvirni znanstveni članek Opis: We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph G, and Alice’s goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round i, where i = 1, 2,, … , the players are taking turns in selecting a previously unselected vertex of G and giving it color i (hence, in each round the selected vertices form an independent set). The game ends when all vertices of G are selected (and thus colored), and the total number of rounds during the game when both players are playing optimally with respect to their goals, is called the independence game chromatic number, χig(G), of G. In fact, four different versions of the independence game chromatic number are considered, which depend on who starts a game and who starts next rounds. We prove that the new invariants lie between the chromatic number of a graph and the maximum degree plus 1, and characterize the graphs in which each of the four versions of the game invariant equals 2. We compare the versions of the independence game chromatic number among themselves and with the classical game chromatic number. In addition, we prove that the independence game chromatic number of a tree can be arbitrarily large.
Ključne besede: graph, coloring, coloring game, competition-independence game, game chromatic number, tree
Objavljeno v DKUM: 09.08.2024; Ogledov: 99; Prenosov: 3
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2. Packings in bipartite prisms and hypercubesBoštjan Brešar, Sandi Klavžar, Douglas F. Rall, 2024, izvirni znanstveni članek Opis: ▫$2$▫-pakirno število ▫$\rho_2(G)$▫ grafa ▫$G$▫ je kardinalnost največjega ▫$2$▫-pakiranja grafa ▫$G$▫, odprto pakirno število ▫$\rho^{\rm o}(G)$▫ pa kardinalnost največjega odprtega pakiranja grafa ▫$G$▫, kjer je odprto pakiranje (oz. ▫$2$▫ pakiranje) množica vozlišč grafa ▫$G$▫, katerih dve (zaprti) soseščini se ne sekata. Dokazano je, da če je ▫$G$▫ dvodelen, potem je ▫$\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$▫. Za hiperkocke sta določeni spodnji meji ▫$\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$▫ in ▫$\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$▫. Te ugotovitve so uporabljene za injektivna barvanja hiperkock. Dokazano je, da je ▫$Q_9$▫ najmanjša hiperkocka, ki ni popolno injektivno obarvljiva. Dokazano je tudi, da je ▫$\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$▫, kjer je ▫$H$▫ poljuben graf brez izoliranih vozlišč.
Ključne besede: 2-pakirno število, odprto pakirno število, dvodelna prizma, hiperkocke, injektivno barvanje, celotno dominacijsko število, 2-packing number, open packing number, bipartite prism, hypercube, injective coloring, total domination number
Objavljeno v DKUM: 28.02.2024; Ogledov: 260; Prenosov: 5
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3. On b-acyclic chromatic number of a graphMarcin Anholcer, Sylwia Cichacz, Iztok Peterin, 2023, izvirni znanstveni članek Opis: Let ▫$G$▫ be a graph. We introduce the acyclic b-chromatic number of ▫$G$▫ as an analog to the b-chromatic number of ▫$G$▫. An acyclic coloring of a graph ▫$G$▫ is a map ▫$c:V(G)\rightarrow \{1,\dots,k\}$▫ such that ▫$c(u)\neq c(v)$▫ for any ▫$uv\in E(G)$▫ and the induced subgraph on vertices of any two colors ▫$i,j\in \{1,\dots,k\}$▫ induce a forest. On a set of all acyclic colorings of a graph ▫$G$▫ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number ▫$A(G)$▫ of ▫$G$▫ and the maximum cardinality of its minimal element is the acyclic b-chromatic number ▫$A_b(G)$▫ of ▫$G$▫. We present several properties of ▫$A_b(G)$▫. In particular, we derive ▫$A_b(G)$▫ for several known graph families, derive some bounds for ▫$A_b(G)$▫, compare ▫$A_b(G)$▫ with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.
Ključne besede: acyclic b-chromatic number, acyclic coloring, b-coloring
Objavljeno v DKUM: 02.08.2023; Ogledov: 421; Prenosov: 14
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4. On acyclic colorings of direct productsSimon Špacapan, Aleksandra Tepeh, 2008, izvirni znanstveni članek Opis: A coloring of a graph ▫$G$▫ is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees ▫$T_1$▫ and ▫$T_2$▫ equals ▫$\min\{ \Delta(T_1) + 1, \Delta(T_2) + 1\}$▫. We also prove that the acyclic chromatic number of direct product of two complete graphs ▫$K_m$▫ and ▫$K_n$▫ is ▫$mn-m-2$▫, where ▫$m \ge n \ge 4$▫. Several bounds for the acyclic chromatic number of direct products are given and in connection to this some questions are raised.
Ključne besede: mathematics, graph theory, coloring, acyclic coloring, distance-two coloring, direct product
Objavljeno v DKUM: 31.03.2017; Ogledov: 909; Prenosov: 127
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5. Explicit homomorphisms of hexagonal graphs to one vertex deleted Petersen graphPetra Šparl, Janez Žerovnik, 2009, izvirni znanstveni članek Opis: Problem odločanja ali obstaja homomorfizem iz poljubnega grafa ▫$G$▫ v dani graf ▫$H$▫ je bil že večkrat proučevan in se je izkazal za zelo težkega. Hell in Nešetril sta dokazala, da je odločitveni problem NP-poln, če ▫$H$▫ ni dvodelen graf. V članku je obravnavan poseben problem, kjer je ▫$G$▫ poljuben heksagonalen graf brez trikotnikov, ▫$H$▫ pa Kneserjev graf ali njegov inducirani podgraf. Podana je esplicitna konstrukcija, ki dokazuje obstoj homomorfizma iz poljubnega heksagonalnega grafa brez trikotnikov v Petersenov graf brez ene točke.
Ključne besede: matematika, teorija grafov, homomorfizem, H-barvanje, heksagonalen graf brez trikotnikov, mathematics, teorija grafov, homomorphism, H-coloring, triangle-free hexagonal graph
Objavljeno v DKUM: 10.07.2015; Ogledov: 1263; Prenosov: 28
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6. Collins, Karen L.(1-WESL-C); Shemmer, Benjamin(1-WESL-C): Constructions of 3-colorable cores. (English summary). - SIAM J. Discrete Math. 16 (2002), no. 1,74--80 (electronic)Sandi Klavžar, 2004, recenzija, prikaz knjige, kritika Ključne besede: matematika, teorija grafov, barvanje, kromatično število, avtomorfizem, mathematics, graph theory, coloring, Chromatic number, automorphisem
Objavljeno v DKUM: 10.07.2015; Ogledov: 1054; Prenosov: 37
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7. 2-local distributed algorithms for generalized coloring of hexagonal graphsPetra Šparl, Janez Žerovnik, 2005, objavljeni znanstveni prispevek na konferenci Opis: A 2-local distributed approximation algorithm for multicoloring of a triangle-free hexagonal graph which uses at most ▫$lceil frac{5omega(G)}{4} rceil + 3$▫ colors is presented.
Ključne besede: matematika, teorija grafov, barvanje grafov, aproksimacijski algoritem, frekvenčni načrt, ▫$k$▫-lokalen porazdeljen algoritem, mathematics, graph theory, approximation algorithms, graph coloring, frequency planning, ▫$k$▫-local distributed algorithm
Objavljeno v DKUM: 10.07.2015; Ogledov: 1285; Prenosov: 100
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8. Simpler multicoloring of triangle-free hexagonal graphsIgnasi Sau Walls, Petra Šparl, Janez Žerovnik, 2012, objavljeni znanstveni prispevek na konferenci Opis: 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).
Ključne besede: matematika, teorija grafov, aproksimacijski algoritem, barvanje grafov, dodeljevanje frekvenc, celične mreže, mathematics, graph algorithm, graph theory, approximation algorithm, graph coloring, frequency planning, cellular networks
Objavljeno v DKUM: 10.07.2015; Ogledov: 1351; Prenosov: 88
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10. How good can ants color graphs?Aleksander Vesel, Janez Žerovnik, 1998 Opis: V notici primerjamo algoritem Coste in Hertza, algoritem "mravlje", s postopkom zaporednega barvanja (RLF, recursive largest first) in z algoritmom tipa Petforf-Welsh. V naših poskusih je zadnji precej boljši od prvih dveh.
Ključne besede: matematika, teorija grafov, barvanje grafov, postopek zaporednega barvanja, algoritem mravlje, Petford-Welsh, RLF, mathematics, graph theory, graph coloring, ants algorithm, Petford-Welsh, RLF
Objavljeno v DKUM: 10.07.2015; Ogledov: 1566; Prenosov: 41
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