scholarly journals Partition of Graphs and Hypergraphs into Monochromatic Connected Parts

10.37236/2121 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Shinya Fujita ◽  
Michitaka Furuya ◽  
András Gyárfás ◽  
Ágnes Tóth
Keyword(s):  
Independence Number ◽  
Uniform Hypergraph ◽  
Colored Graph ◽  
Edge Colorings ◽  
Colored Graphs ◽  
Vertex Set ◽  
Special Case

We show that two results on covering of edge colored graphs by monochromatic connected parts can be extended to partitioning. We prove that for any $2$-edge-colored non-trivial $r$-uniform hypergraph $H$, the vertex set can be partitioned into at most $\alpha (H)-r+2$ monochromatic connected parts, where $\alpha (H)$ is the maximum number of vertices that does not contain any edge. In particular, any $2$-edge-colored graph $G$ can be partitioned into $\alpha(G)$ monochromatic connected parts, where $\alpha (G)$ denotes the independence number of $G$. This extends König's theorem, a special case of Ryser's conjecture. Our second result is about Gallai-colorings, i.e. edge-colorings of graphs without $3$-edge-colored triangles. We show that for any Gallai-coloring of a graph $G$, the vertex set of $G$ can be partitioned into monochromatic connected parts, where the number of parts depends only on $\alpha(G)$. This extends its cover-version proved earlier by Simonyi and two of the authors.

scholarly journals Monochromatic Loose-Cycle Partitions in Hypergraphs

10.37236/4062 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
András Gyárfás ◽  
Gábor Sárközy
Keyword(s):  
Independence Number ◽  
Uniform Hypergraph ◽  
Partition Problem ◽  
Vertex Set

In this paper we study the monochromatic loose-cycle partition problem for non-complete hypergraphs. Our main result is that in any $r$-coloring of a $k$-uniform hypergraph with independence number $\alpha$ there is a partition of the vertex set into monochromatic loose cycles such that their number depends only on $r$, $k$ and $\alpha$. We also give an extension of the following result of Pósa to hypergraphs: the vertex set of every graph $G$ can be partitioned into at most $\alpha(G)$ cycles, edges and vertices.

scholarly journals Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

10.37236/773 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jeremy F. Alm ◽  
Roger D. Maddux ◽  
Jacob Manske
Keyword(s):  
Complete Graph ◽  
Ramsey Theory ◽  
Projective Planes ◽  
Relation Algebras ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
Finite Set ◽  
Vertex Set ◽  
Special Case

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.

scholarly journals Turán Density of $2$-Edge-Colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shuliang Bai ◽  
Linyuan Lu
Keyword(s):  
Bipartite Graphs ◽  
Important Application ◽  
Edge Density ◽  
Colored Graph ◽  
Colored Graphs ◽  
Turan Problems ◽  
Vertex Set

We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ where $E_r$ and $E_b$ do not have to be disjoint. The Turán density $\pi(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/\binom{n}{2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application on the Turán problems of $\{2, 3\}$-hypergraphs.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

scholarly journals Monochromatic Path and Cycle Partitions in Hypergraphs

10.37236/2631 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
András Gyárfás ◽  
Gábor N. Sárközy
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

scholarly journals On paths, trails and closed trails in edge-colored graphs

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Laurent Gourvès ◽  
Adria Lyra ◽  
Carlos A. Martinhon ◽  
Jérôme Monnot
Keyword(s):  
Polynomial Time ◽  
Optimal Solution ◽  
Colored Graph ◽  
Edge Disjoint ◽  
Colored Graphs ◽  
International Audience ◽  
Np Complete ◽  
T Path ◽  
Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

scholarly journals A Note on the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

10.37236/7329 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
Keyword(s):  
Independent Set ◽  
Uniform Hypergraph ◽  
Edge Disjoint ◽  
Cycle Cover ◽  
Vertex Set ◽  
Cyclic Sequence

A linear cycle in a $3$-uniform hypergraph $H$ is a cyclic sequence of hyperedges such that any two consecutive hyperedges intersect in exactly one element and non-consecutive hyperedges are disjoint. Let $\alpha(H)$ denote the size of a largest independent set of $H$.We show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha(H)$ edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.

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