scholarly journals Avoidable Paths in Graphs

10.37236/9030 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Marthe Bonamy ◽  
Oscar Defrain ◽  
Meike Hatzel ◽  
Jocelyn Thiebaut
Keyword(s):  
Positive Integer ◽  
Induction Hypothesis ◽  
Elementary Proof ◽  
Chordal Graphs

We prove a recent conjecture of Beisegel et al. that for every positive integer $k$, every graph containing an induced $P_k$ also contains an avoidable $P_k$. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for $k \in \{1,2\}$ (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions.  We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.

scholarly journals Open k-monopolies in graphs: complexity and related concepts

2016 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael G. Yero
Keyword(s):  
Positive Integer ◽  
Computer Science ◽  
Long Range ◽  
Theoretical Computer Science ◽  
Chordal Graphs ◽  
Discrete Mathematics ◽  
Voting Systems ◽  
Minimum Cardinality ◽  
Theoretical Computer ◽  
Np Complete

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values. Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016)

On a Theorem of Rav Concerning Egyptian Fractions

1975 ◽  
Vol 18 (1) ◽  
pp. 155-156 ◽  
Author(s):  
William A. Webb
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Egyptian Fractions ◽  
Image Position ◽  
Complete Bibliography ◽  
Necessary And Sufficient

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.

scholarly journals An Extension of a Congruence by Tauraso

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Romeo Meštrović
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Harmonic Number ◽  
Combinatorial Identity ◽  
Harmonic Numbers

For a positive integer let be the th harmonic number. In this paper we prove that, for any prime ,  . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.

scholarly journals On the Complexity of Subfall Coloring of Graphs

2021 ◽  
Author(s):  
Davi De Andrade ◽  
Ana Silva
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Chordal Graphs ◽  
Seminal Paper ◽  
Color Class ◽  
Np Complete

Given a graph G and a proper k-coloring f of G, a b-vertex in f is a vertex that is adjacent to every color class but its own. If every vertex is a b-vertex, then f is a fall k-coloring; and G has a subfall k-coloring if there is an induced H $\subseteq$ G such that H has a fall k-coloring. The subfall chromatic number of G is the largest positive integer $\psi_{fs}(G)$ such that G has a subfall $\psi_{fs}(G)$-coloring. We prove that deciding if a graph G has a subfall k-coloring is NP-complete for k $\geq$ 4, and characterize graphs having a subfall 3-coloring. This answers a question posed by Dunbar et al. (2000) in their seminal paper. We also prove polinomiality for chordal graphs and cographs, and present a comparison with other known coloring metrics based on heuristics.

Spectra of independent power measures

1972 ◽  
Vol 72 (1) ◽  
pp. 27-35 ◽  
Author(s):  
W. J. Bailey ◽  
G. Brown ◽  
W. Moran
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Elementary Proof ◽  
Natural Generalization ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Power Measure ◽  
Lca Groups

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

scholarly journals An elementary proof of Fermat’s last theorem for all even exponents

2020 ◽  
Vol 14 (1) ◽  
pp. 139-142
Author(s):  
Sudhangshu B. Karmakar
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Integer Solutions

AbstractAn elementary proof that the equation x2n + y2n = z2n can not have any non-zero positive integer solutions when n is an integer ≥ 2 is presented. To prove that the equation has no integer solutions it is first hypothesized that the equation has integer solutions. The absence of any integer solutions of the equation is justified by contradicting the hypothesis.

scholarly journals A Theorem of Fermat on Congruent Number Curves

2019 ◽  
Author(s):  
Lorenz Halbeisen ◽  
Norbert Hungerbühler
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Elementary Proof ◽  
Rational Point ◽  
Rational Points ◽  
Congruent Number ◽  
International Audience ◽  
Number Curve

International audience A positive integer $A$ is called a \emph{congruent number} if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a \emph{congruent number} if and only if the congruent number curve $y^2 = x^3 − A^2 x$ has a rational point $(x, y) \in {\mathbb{Q}}^2$ with $y \ne 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.

scholarly journals Powers of chordal graphs

1983 ◽  
Vol 35 (2) ◽  
pp. 211-217 ◽  
Author(s):  
R. Balakrishnan ◽  
P. Paulraja
Keyword(s):  
Positive Integer ◽  
Chordal Graph ◽  
Simple Graph ◽  
Chordal Graphs ◽  
Chordless Cycle

AbstractAn undirected simple graph G is called chordal if every circle of G of length greater than 3 has a chord. For a chordal graph G, we prove the following: (i) If m is an odd positive integer, Gm is chordal. (ii) If m is an even positive integer and if Gm is not chordal, then none of the edges of any chordless cycle of Gm is an edge of Gr, r < m.

scholarly journals An elementary proof of some character sum identities of Apostol

1973 ◽  
Vol 14 (1) ◽  
pp. 50-53 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Residue Class ◽  
Character Sum ◽  
Primitive Character ◽  
Image Position

Let х denote a primitive character modulo k. Using two different representations for Dirichlet L-functions, Apostol [1] recently derived a representation forinvolving the sumswhere m is a positive integer. Furthermore, if х(r) = (r|p)the residue class character modulo the odd prime p, he derived a representation for Mm(х) involving the sums

scholarly journals Bipartite powers of k-chordal graphs

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Sunil Chandran ◽  
Rogers Mathew
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Discrete Mathematics ◽  
International Audience ◽  
Positive Integers ◽  
Chordal Bipartite Graphs

Graph Theory International audience Let k be an integer and k ≥3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if Gm is chordal then so is Gm+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if Gm is k-chordal, then so is Gm+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m-th bipartite power G[m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u,v) where dG(u,v) is odd and less than or equal to m. Note that G[m] = G[m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G[m], where k, m are positive integers with k≥4.

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