Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.Coloring problem of signed interval graphs
http://toc.ui.ac.ir/article_23849_4289.html
A signed graph $(G,sigma)$ is a graph‎ ‎together with an assignment of signs ${+,-}$ to its edges where‎ ‎$sigma$ is the subset of its negative edges‎. ‎There are a few variants of coloring and clique problems of‎ ‎signed graphs‎, ‎which have been studied‎. ‎An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$‎. ‎Recently Naserasr et. al., in [R‎. ‎Naserasr‎, ‎E‎. ‎Rollova and E‎. ‎Sopena‎, ‎Homomorphisms of signed graphs‎, ‎J‎. ‎Graph Theory‎, 79‎‎ (2015) 178--212, have defined signed chromatic and signed clique numbers of signed graphs‎. ‎In this paper we consider the latter mentioned problems for signed interval graphs‎. ‎We prove that the coloring problem of signed‎ ‎interval graphs is NP-complete whereas their ordinary coloring‎ ‎problem (the coloring problem of interval graphs) is in P‎. ‎Moreover we prove that the signed clique problem of a‎ ‎signed interval graph can be solved in polynomial time‎. ‎We also consider the‎ ‎complexity of further related problems‎. Sat, 30 Nov 2019 20:30:00 +0100Elliptic root systems of type $A_1$, a combinatorial study
http://toc.ui.ac.ir/article_24023_4289.html
We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.Sat, 30 Nov 2019 20:30:00 +0100Generalized zagreb index of product graphs
http://toc.ui.ac.ir/article_24024_0.html
The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices.In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.Thu, 10 Oct 2019 20:30:00 +0100Bounds for metric dimension and defensive k-alliance of graphs under deleted lexicographic product
http://toc.ui.ac.ir/article_24081_0.html
Metric dimension and defensive k-alliance number are two distance-based graphinvariants which have applications in robot navigation, quantitative analysis ofsecondary RNA structures, national defense and fault-tolerant computing. In thispaper, some bounds for metric dimension and defensive k-alliance of deleted lexicographicproduct of graphs are presented. We also show that the bounds aresharp.Mon, 28 Oct 2019 20:30:00 +0100Some subgroups of $\mathbb{F}_q^*$ and explicit factors of $x^{2^nd}-1 \in \mathbb{F}_q[x]$
http://toc.ui.ac.ir/article_24139_0.html
Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$. Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$. In this paper, we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and, $mathcal{O}_q=langle trangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. Further, we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$.Sat, 09 Nov 2019 20:30:00 +0100A Linear Algorithm for Computing $gamma_{[1,2]}$-set in Generalized Series-Parallel Graphs
http://toc.ui.ac.ir/article_24185_0.html
For a graph $G=(V,E)$, a set $S subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v in V setminus S$ is dominated by at most two vertices of $S$, i.e. $1 leq vert N(v) cap S vert leq 2$.Moreover a set $S subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 leq vert N(v) cap S vert leq 2$. The $[1,2]$-domination number of $G$, denoted $gamma_{[1,2]}(G)$,is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $gamma_{[1,2]}(G)$ is called a $gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $gamma_{[1,2]}$-set and a $gamma_{t[1,2]}$-set in generalized series-parallel graphs.Sat, 23 Nov 2019 20:30:00 +0100