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51.
A note on the chromatic number of the square of the Cartesian product of two cycles
Zehui Shao, Aleksander Vesel, 2013, kratki znanstveni prispevek

Opis: The square ▫$G^2$▫ of a graph ▫$G$▫ is obtained from ▫$G$▫ by adding edges joining all pairs of nodes at distance 2 in ▫$G$▫. In this note we prove that ▫$chi((C_mBox C_n)^2) le 6$ for $m, n ge 40$▫. This confirms Conjecture 19 stated in [É. Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327-2333].
Ključne besede: matematika, teorija grafov, kromatično število, kartezični produkt, označevanje grafov, kvadrat grafa, mathematics, graph theory, chromatic number, Cartesian product, graph labeling, square if a graph
Objavljeno: 10.07.2015; Ogledov: 486; Prenosov: 43
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52.
On Vizing's conjecture
Boštjan Brešar, 2001, izvirni znanstveni članek

Opis: A dominating set ▫$D$▫ gor a graph ▫$G$▫ is a subset ▫$V(G)$▫ such that any vertex in ▫$V(G)-D$▫ has a neighbor in ▫$D$▫, and a domination number ▫$\gamma(G)$▫ is the size of a minimum dominating set for ▫$G$▫. For the Cartesian product ▫$G \Box H$▫ Vizing's conjecture states that ▫$\gamma(G \Box H) \ge \gamma(G)\gamma(H)$▫ for every pair of graphs ▫$G,H$▫. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when ▫$\gamma(G) = \gamma(H) = 3$▫.
Ključne besede: mathematics, graph theory, graph, Cartesian product, domination number
Objavljeno: 31.03.2017; Ogledov: 432; Prenosov: 47
.pdf Celotno besedilo (126,98 KB)
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53.
Computing the weighted Wiener and Szeged number on weighted cactus graphs in linear time
Blaž Zmazek, Janez Žerovnik, 2003, izvirni znanstveni članek

Opis: Cactus is a graph in which every edge lies on at most one cycle. Linear algorithms for computing the weighted Wiener and Szeged numbers on weighted cactus graphs are given. Graphs with weighted vertices and edges correspond to molecular graphs with heteroatoms.
Ključne besede: mathematics, graph theory, Wiener number, Szeged number, weighted cactus, linear algorithm
Objavljeno: 05.07.2017; Ogledov: 178; Prenosov: 41
.pdf Celotno besedilo (130,41 KB)
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54.
Wiener numbers of pericondensed benzenoid hydrocarbons
Sandi Klavžar, Ivan Gutman, Amal Rajapakse, 1997, izvirni znanstveni članek

Opis: Using a recently developed technique for the calculation of the Wiener number (W) of benzenoid systems, we determine explicit expressions for W for several homologous series of pericondensed benzenoid hydrocarbons. An elementary proof for the correctness of the used method is also included.
Ključne besede: mathematics, chemical graph theory, distance in graphs, Wiener number, benzenoids
Objavljeno: 05.07.2017; Ogledov: 311; Prenosov: 41
.pdf Celotno besedilo (16,93 MB)
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55.
Roman domination number of the Cartesian products of paths and cycles
Polona Repolusk, Janez Žerovnik, 2012, izvirni znanstveni članek

Opis: Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from $\{0,1,2\}$. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give ▫$O(C)$▫ time algorithm for computing Roman domination number of special classes of polygraphs (rota- and fasciagraphs). By implementing the algorithm we give formulas for Roman domination number of the Cartesian products of paths and cycles ▫$P_n \Box P_k$▫, ▫$P_n \Box C_k$▫ for ▫$k \leq 8$▫ and ▫$n \in {\mathbb N}$▫ and for ▫$C_n \Box P_k$▫ and ▫$C_n \Box C_k$▫ for ▫$k \leq 5$▫, ▫$n \in {\mathbb N}$▫. We also give a list of Roman graphs among investigated families.
Ključne besede: graph theory, Roman domination number, Cartesian product, polygraphs, path algebra
Objavljeno: 23.08.2017; Ogledov: 249; Prenosov: 35
.pdf Celotno besedilo (719,06 KB)
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