scholarly journals Distinguishing Chromatic Numbers of Bipartite Graphs

10.37236/165 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Laflamme ◽  
K. Seyffarth
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Chromatic Numbers

Extending the work of K.L. Collins and A.N. Trenk, we characterize connected bipartite graphs with large distinguishing chromatic number. In particular, if $G$ is a connected bipartite graph with maximum degree $\Delta \geq 3$, then $\chi_D(G)\leq 2\Delta -2$ whenever $G\not\cong K_{\Delta-1,\Delta}$, $K_{\Delta,\Delta}$.

Algorithmic Aspects of Partial List Colourings

2000 ◽  
Vol 9 (4) ◽  
pp. 375-380 ◽  
Author(s):  
M. VOIGT
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Polynomial Time ◽  
Bipartite Graphs ◽  
Partial List ◽  
List Chromatic Number

Let G = (V, E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χ[lscr ](G). Suppose each vertex of V(G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least [formula here] vertices can be coloured properly from these lists.Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 [les ] t [les ] χ[lscr ](G), it is possible to colour at least (1 − (1/2)t)n vertices in polynomial time. Thus, if G is bipartite and [Lscr ] is a list assignment with [mid ]L(v)[mid ] [ges ] log2n for all v ∈ V, then G is [Lscr ]-list colourable in polynomial time.

scholarly journals On star coloring of Mycielskians

2018 ◽  
Vol 2 (2) ◽  
pp. 82
Author(s):  
K. Kaliraj ◽  
V. Kowsalya ◽  
Vernold Vivin
Keyword(s):  
Chromatic Number ◽  
Graph Transformation ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Complete Bipartite Graphs ◽  
Star Coloring

<p>In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph <span class="math"><em>G</em></span> into a new graph <span class="math"><em>μ</em>(<em>G</em>)</span>, we now call the Mycielskian of <span class="math"><em>G</em></span>, which has the same clique number as <span class="math"><em>G</em></span> and whose chromatic number equals <span class="math"><em>χ</em>(<em>G</em>) + 1</span>. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.</p>

scholarly journals Results on a Chromatic Number of a Bi-polar Fuzzy Complete Bipartite Graphs and Labelling of Tri-Polar Fuzzy Graphs

2021 ◽  
Vol 12 (2) ◽  
pp. 1040-1046
Author(s):  
Remala Mounika Lakshmi, Et. al.
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Bipartite Graph ◽  
Research Work ◽  
Bipartite Graphs ◽  
Network System ◽  
Route Network ◽  
Fuzzy Graphs ◽  
Complete Bipartite Graphs ◽  
Ultimate Objective

The ultimate objective of a piece of research work is to present the labelling of vertices in 3-PFG and labelling of distances in 3-PFG. Also, we characterize some of its properties. Later, we define the vertex and edge chromatic number BF- Complete Bipartite graph. Further we illustrated an example for BFRGS which represents a Route Network system.

scholarly journals Spanning $k$-trees of Bipartite Graphs

10.37236/3628 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mikio Kano ◽  
Kenta Ozeki ◽  
Kazuhiro Suzuki ◽  
Masao Tsugaki ◽  
Tomoki Yamashita
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Sum

A tree is called a $k$-tree if its maximum degree is at most $k$. We prove the following theorem. Let $k \geq 2$ be an integer, and $G$ be a connected bipartite graph with bipartition $(A,B)$ such that $|A| \le |B| \le (k-1)|A|+1$. If $\sigma_k(G) \ge |B|$, then $G$ has a spanning $k$-tree, where $\sigma_k(G)$ denotes the minimum degree sum of $k$ independent vertices of $G$. Moreover, the condition on $\sigma_k(G)$ is sharp. It was shown by Win (Abh. Math. Sem. Univ. Hamburg, 43, 263–267, 1975) that if a connected graph $H$ satisfies $\sigma_k(H) \ge |H|-1$, then $H$ has a spanning $k$-tree. Thus our theorem shows that the condition becomes much weaker if the graph is bipartite.

scholarly journals The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph

2020 ◽  
Vol 30 (03) ◽  
pp. 2040005
Author(s):  
Yingzhi Tian ◽  
Huaping Ma ◽  
Liyun Wu
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Type Inequality ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Complementary Graph

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

scholarly journals Extensions to 2-Factors in Bipartite Graphs

10.37236/3594 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jennifer Vandenbussche ◽  
Douglas B. West
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Bipartite Graphs

A graph is $d$-bounded if its maximum degree is at most $d$.  We apply the Ore-Ryser Theorem on $f$-factors in bipartite graphs to obtain conditions for the extension of a $2$-bounded subgraph to a $2$-factor in a bipartite graph.  As consequences, we prove that every matching in the $5$-dimensional hypercube extends to a $2$-factor, and we obtain conditions for this property in general regular bipartite graphs.  For example, to show that every matching in a regular $n$-vertex bipartite graph extends to a $2$-factor, it suffices to show that all matchings with fewer than $n/3$ edges extend to $2$-factors.

scholarly journals Bipartite Graphs whose Squares are not Chromatic-Choosable

10.37236/4343 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Boram Park
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Total Coloring ◽  
Natural Question ◽  
List Total Coloring ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
List Chromatic Number ◽  
Total Coloring Conjecture

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called chromatic-choosable if $\chi_{\ell} (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable.Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs and are not chromatic choosable. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$.In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2) \neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) - \chi(G^2)$ can be arbitrarily large.

scholarly journals Partitioning to three matchings of given size is NP-complete for bipartite graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 206-209 ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Perfect Matching ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Np Complete

Abstract We show that the problem of deciding whether the edge set of a bipartite graph can be partitioned into three matchings, of size k1, k2 and k3 is NP-complete, even if one of the matchings is required to be perfect. We also show that the problem of deciding whether the edge set of a simple graph contains a perfect matching and a disjoint matching of size k or not is NP-complete, already for bipartite graphs with maximum degree 3. It also follows from our construction that it is NP-complete to decide whether in a bipartite graph there is a perfect matching and a disjoint matching that covers all vertices whose degree is at least 2.

scholarly journals Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

10.37236/5512 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Vladimir Blinovsky ◽  
Catherine Greenhill
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Asymptotic Enumeration ◽  
Uniform Hypergraphs ◽  
Enumeration Formula ◽  
Linear Hypergraphs

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

scholarly journals On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

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