ON THE CHROMATIC NUMBERS OF FUNCTIGRAPHS

2012 ◽  
Vol 13 (03n04) ◽  
pp. 1250011 ◽  
Author(s):  
GEORGE QI ◽  
SHENGHAO WANG ◽  
WEIZHEN GU
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Sufficient Conditions ◽  
Chromatic Numbers ◽  
Disjoint Copy ◽  
Minimum Number ◽  
Definition Of

The chromatic number of a graph G, denoted χ(G) is the minimum number of colors needed to color vertices of G so that no two adjacent vertices share the same color. A functigraph over a given graph is obtained as follows: Let G' be a disjoint copy of a given G and f be a function f : V(G) → V(G'). The functigraph over G, denoted by C(G, f), is the graph with V(C(G, f)) = V(G) ∪ V(G') and E(C(G, f)) = E(G) ∪ E(G') ∪ {uv : u ∈ V(G), v ∈ V(G'), v = f(u)}. Recently, Chen et al. proved that [Formula: see text]. In this paper, we first provide sufficient conditions on functions f to reach the lower bound for any graph. We then study the attainability of the chromatic numbers of functigraphs. Finally, we extend the definition of a functigraph in different ways and then investigate the bounds of chromatic numbers of such graphs.

scholarly journals The local antimagic on disjoint union of some family graphs

2019 ◽  
Vol 5 (2) ◽  
pp. 69-75
Author(s):  
Marsidi Marsidi ◽  
Ika Hesti Agustin
Keyword(s):  
Lower Bound ◽  
Natural Number ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Undirected Graph ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
Vertex Set ◽  
Proper Vertex ◽  
Vertex Colouring

A graph  in this paper is nontrivial, finite, connected, simple, and undirected. Graph  consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of  is not equal with the weight of , where the weight of  (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where  is the colour on the vertex . The vertex local antimagic chromatic number  is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

scholarly journals Local chromatic number and topology

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gábor Simonyi ◽  
Gábor Tardos
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Topological Method ◽  
Proper Coloring ◽  
And Topology ◽  
Minimum Number ◽  
International Audience ◽  
Closed Neighborhood

International audience The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.

scholarly journals Colorful Subhypergraphs in Kneser Hypergraphs

10.37236/3573 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Frédéric Meunier
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Complete Bipartite Graphs ◽  
Definition Of ◽  
Ky Fan

Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

scholarly journals Acyclic, Star and Oriented Colourings of Graph Subdivisions

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Undirected Graph ◽  
The Other ◽  
Chromatic Index ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
International Audience ◽  
Vertex Partitions ◽  
Colour Classes ◽  
Vertex Colouring

International audience Let G be a graph with chromatic number χ (G). A vertex colouring of G is \emphacyclic if each bichromatic subgraph is a forest. A \emphstar colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ _a(G) and χ _s(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ _a(G'), χ _s(G') and χ (G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The \emphoriented chromatic number χ ^→(G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ ^→(G')=χ (G) whenever χ (G)≥ 9.

scholarly journals An Inertial Lower Bound for the Chromatic Number of a Graph

10.37236/6404 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Clive Elphick ◽  
Pawel Wocjan
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Extremal Graphs ◽  
Chromatic Numbers

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that:\[1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G)\] and conjecture that \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G).\] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.

scholarly journals LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS

2021 ◽  
Vol 55 (2 (255)) ◽  
pp. 96-112
Author(s):  
Aram H. Gharibyan ◽  
Petros A. Petrosyan
Keyword(s):  
Chromatic Number ◽  
Open Neighborhood ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
Closed Neighborhood

In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as  follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$.  A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balanced with a closed  neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number $k$ ($k\geq 2$), for which a graph $G$ has a locally-balanced $k$-partition with an open (a closed) neighborhood, is called an         $lb$-open ($lb$-closed) chromatic number of $G$ and denoted by                   $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the $lb$-open and $lb$-closed chromatic numbers of several families of graphs. We also consider the connections of $lb$-open and $lb$-closed chromatic numbers of graphs with other chromatic numbers such as injective and $2$-distance chromatic numbers.

A Philosophical Analysis To Uncover The Meaning And Terminology Of Person In Indonesian Criminal Law Context

2017 ◽  
Vol 1 (1) ◽  
pp. 56
Author(s):  
Nani Mulyati ◽  
Topo Santoso ◽  
Elwi Danil
Keyword(s):  
Criminal Law ◽  
Sufficient Conditions ◽  
Criminal Code ◽  
Legal Person ◽  
Philosophical Approach ◽  
Legal Personality ◽  
Legal Subject ◽  
Long Time ◽  
Definition Of ◽  
Broad Meaning

The definition of person and non-person always change through legal history. Long time ago, law did not recognize the personality of slaves. Recently, it accepted non-human legal subject as legitimate person before the law. This article examines sufficient conditions for being person in the eye of law according to its particular purposes, and then, analyses the meaning of legal person in criminal law. In order to do that, scientific methodology that is adopted in this research is doctrinal legal research combined with philosophical approach. Some theories regarding person and legal person were analysed, and then the concept of person was associated with the accepted definition of legal person that is adopted in the latest Indonesian drafted criminal code. From the study that has been done, can be construed that person in criminal law concerned with norm adressat of the rule, as the author of the acts or omissions, and not merely the holder of rights. It has to be someone or something with the ability to think rationally and the ability to be responsible for the choices he/she made. Drafted penal code embraces human and corporation as its norm adressat. Corporation defined with broad meaning of collectives. Consequently, it will include not only entities with legal personality, but also associations without legal personality. Furthermore, it may also hold all kind of collective namely states, states bodies, political parties, state’s corporation, be criminally liable.

scholarly journals Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 730
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Comparison Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

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