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

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.

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Partition of Graphs and Hypergraphs into Monochromatic Connected Parts

The Electronic Journal of Combinatorics ◽

10.37236/2121 ◽

2012 ◽

Vol 19 (3) ◽

Cited By ~ 2

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.

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Cycle partitions of regular graphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000553 ◽

2020 ◽

pp. 1-24

Author(s):

Vytautas Gruslys ◽

Shoham Letzter

Keyword(s):

Regular Graph ◽

Bipartite Graphs ◽

Disjoint Paths ◽

Regular Graphs ◽

Simple Construction ◽

Vertex Set ◽

Almost All ◽

Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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Bipartite graphs with close domination and k-domination numbers

Open Mathematics ◽

10.1515/math-2020-0047 ◽

2020 ◽

Vol 18 (1) ◽

pp. 873-885

Author(s):

Gülnaz Boruzanlı Ekinci ◽

Csilla Bujtás

Keyword(s):

Positive Integer ◽

Dominating Set ◽

Domination Number ◽

Bipartite Graphs ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Minimum Cardinality ◽

Vertex Set ◽

Necessary And Sufficient ◽

The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

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.

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On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽

10.3390/math8101778 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1778

Author(s):

Fangyun Tao ◽

Ting Jin ◽

Yiyou Tu

Keyword(s):

Upper Bound ◽

Maximum Degree ◽

Bipartite Graphs ◽

Equitable Partition ◽

Special Cases ◽

Vertex Set ◽

Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

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Rainbow Turán Problems for Paths and Forests of Stars

The Electronic Journal of Combinatorics ◽

10.37236/6430 ◽

2017 ◽

Vol 24 (1) ◽

Author(s):

Daniel Johnston ◽

Cory Palmer ◽

Amites Sarkar

Keyword(s):

Systematic Study ◽

Colored Graph ◽

Turán Number ◽

Turan Problems ◽

Different Color

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Turán number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

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Genericity of chaos for colored graphs

Topological Algebra and its Applications ◽

10.1515/taa-2020-0105 ◽

2021 ◽

Vol 9 (1) ◽

pp. 37-52

Author(s):

Ramón Barral Lijó ◽

Hiraku Nozawa

Keyword(s):

Colored Graph ◽

Isomorphism Classes ◽

Colored Graphs ◽

Universal Space

Abstract To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney’s. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.

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A Note on the Speed of Hereditary Graph Properties

The Electronic Journal of Combinatorics ◽

10.37236/644 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 4

Author(s):

Vadim V. Lozin ◽

Colin Mayhill ◽

Victor Zamaraev

Keyword(s):

Practical Importance ◽

Bipartite Graphs ◽

Hereditary Property ◽

Induced Subgraphs ◽

Forbidden Induced Subgraphs ◽

Graph Properties ◽

Graph Property ◽

Vertex Set ◽

Split Graphs ◽

Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

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A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices

Opuscula Mathematica ◽

10.7494/opmath.2020.40.3.375 ◽

2020 ◽

Vol 40 (3) ◽

pp. 375-382

Author(s):

Narges Ghareghani ◽

Iztok Peterin ◽

Pouyeh Sharifani

Keyword(s):

Decision Problem ◽

Dominating Set ◽

Domination Number ◽

Short Note ◽

Bipartite Graphs ◽

Hard Problem ◽

Simple Construction ◽

Minimum Number ◽

Vertex Set ◽

Np Hard Problem

A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of vertices is called a $\gamma_{[1,k]}$-set and the number of its vertices is the $[1,k]$-domination number $\gamma_{[1,k]}(G)$ of $G$. In this short note we show that the decision problem whether $\gamma_{[1,k]}(G)=n$ is an $NP$-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph $G$ of order $n$ satisfying $\gamma_{[1,k]}(G)=n$ is given for every integer $n \geq (k+1)(2k+3)$.

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On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

Journal of Interconnection Networks ◽

10.1142/s0219265920500073 ◽

2020 ◽

Vol 20 (02) ◽

pp. 2050007

Author(s):

P. C. LISNA ◽

M. S. SUNITHA

Keyword(s):

Image Processing ◽

Chromatic Number ◽

Bipartite Graphs ◽

Complete Graphs ◽

Proper Coloring ◽

Color Class ◽

Vertex Set ◽

Chromatic Sum ◽

Complete Bipartite Graphs

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

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