Strong Chromatic Number Research Articles

Packing list‐colorings

AbstractList coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list‐coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a ‐list‐assignment of a graph , which is the assignment of a list of colors to each vertex , we study the existence of pairwise‐disjoint proper colorings of using colors from these lists. We may refer to this as a list‐packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest for which such a list‐packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of . (The reader might already find it interesting that such a minimal is well defined.) We also pursue a more focused study of the case when is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalizations of the problem above in the same spirit.

Strong Vertex Coloring in Bipolar Fuzzy Graphs

Bipolar fuzzy graph (BFG) coloring techniques are used to solve many complex real world problems. The chromatic number of complement of BFG is obtained and compared with the chromatic number of the corresponding BFGs. This paper is an attempt to define coloring in a BFG based on strong edges. The strong chromatic number of complete BFG and BF tree are obtained.

On the Strong Chromatic Number of Random Hypergraphs

On the Strong Chromatic Number of Random Hypergraphs

Algorithms for Weighted Independent Transversals and Strong Colouring

An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.

Colouring of COVID-19 Affected Region Based on Fuzzy Directed Graphs

Graph colouring is the system of assigning a colour to each vertex of a graph. It is done in such a way that adjacent vertices do not have equal colour. It is fundamental in graph theory. It is often used to solve real-world problems like traffic light signalling, map colouring, scheduling, etc. Nowadays, social networks are prevalent systems in our life. Here, the users are considered as vertices, and their connections/interactions are taken as edges. Some users follow other popular users’ profiles in these networks, and some don’t, but those non-followers are connected directly to the popular profiles. That means, along with traditional relationship (information flowing), there is another relation among them. It depends on the domination of the relationship between the nodes. This type of situation can be modelled as a directed fuzzy graph. In the colouring of fuzzy graph theory, edge membership plays a vital role. Edge membership is a representation of flowing information between end nodes of the edge. Apart from the communication relationship, there may be some other factors like domination in relation. This influence of power is captured here. In this article, the colouring of directed fuzzy graphs is defined based on the influence of relationship. Along with this, the chromatic number and strong chromatic number are provided, and related properties are investigated. An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs.

On strict strong coloring of graphs

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

On the strong chromatic number of a random 3-uniform hypergraph

This paper deals with estimating the threshold for the strong r-colorability of a random 3-uniform hypergraph in the binomial model H(n,3,p). A vertex coloring is said to be strong for a hypergraph if every two vertices sharing a common edge are colored with distinct colors. It is known that the threshold corresponds to the sparse case, when the expected number of edges is a linear function of n, pn3=cn, and c>0 depends on r, but not on n. We establish the threshold as a bound on the parameter c up to an additive constant. In particular, by using the second moment method we prove that for large enough r and c<rlnr3−518lnr−13−r−1∕6, the random hypergraph H(n,3,p) is strongly r-colorable with high probability and, vice versa, for c>rlnr3−518lnr+O(lnr∕r), it is not strongly r-colorable with high probability.

Edge Balanced 3-Uniform Hypergraph Designs

In this paper, we completely determine the spectrum of edge balanced H-designs, where H is a 3-uniform hypergraph with 2 or 3 edges, such that H has strong chromatic number χs(H)=3.

An asymptotic bound for the strong chromatic number

AbstractThe strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c &gt; 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

The Wireless Sensor Networks Based on Adjacent Strong Edge Chromatic Number

The adjacent strong edge chromatic number of graphs can be applied to the optimization design of wireless sensor network to make the network design more reasonable. An algorithm,based on the basic principle of page sorting technology, is designed to search for the adjacent strong edge coloring of certain join graphs. Therefore, the adjacent strong edge chromatic number of a join graph is obtained, and it can avoid the repetition of wireless sensor network.

On the Equitable Adjacent Strong Edge Coloring of K(n,m)

For a graph G(V,E),then is called k-adjacent strong edge coloring of G, is abbreviated k-ASEC, if a k-ASEC of G satisfies then f is called k-equitable adjacent strong edge coloring of G, is called k-EASC for short, and is called the equitable adjacent strong edge chromatic number of G. In this paper, we have proved the equitable adjacent strong edge chromatic number of graph K(n,m) , with n≥ 2,n≡0(mod 2), m≥ 1.

"'k,r' Semi Strong Chromatic Number of a Graph"

-semi strong chromatic number of G and it is denoted by (G). The number (G) is determined for various known graphs and some bounds are obtained for it . -chromatic number.

The adjacent strong edge chromatic number of P&lt;SUB align=right&gt;1 &amp;or; K&lt;SUB align=right&gt;m,n C&lt;SUB align=right&gt;2 &amp;or; K&lt;SUB align=right&gt;m,n

An algorithm is designed to search for the k-adjacent strong edge colouring of certain join graphs. Therefore, the adjacent strong edge chromatic number of two join graphs is obtained.

Strong Local Colorings of Coronas

In this paper, we study strong local colorings of some important families of coronas. A local coloring of a graph G of order at least 2 is a function $c : V (G) \to N$ such that for every set $S\subseteq V (G)$ with $2\le |S| \le 3,$ there exists two distinct vertices $u,v\in S$ such that $|c(u)-c(v)|\ge m_s,$ where $m_s$ is the size of the induced subgraph . The value of a local coloring c is the maximum color it assigns to a vertex of G. The local chromatic number of $G$ is the minimum value of any local coloring of G and we denote it by $\chi_\ell(G).$ A local coloring of $G$ with value $\chi_{\ell}(G)$ is called a minimum local coloring of G. If a minimum local coloring of G uses all the $\chi_\ell(G)$ colors then it is called a strong local coloring of G. If every minimum local coloring of G uses all the $\chi_\ell(G)$ colors then G is called strong local colorable and in this case, its local chromatic number is called strong local chromatic number and is denoted by $\chi_{s\ell}(G).$ In this paper, we have considered some important families of coronas and determined the strong local chromatic number, if it exists; otherwise, we have proved that they are not strong local colorable but local colorable and determined their local chromatic number. Keywords: Local coloring; strong local coloring; local chromatic number; strong local chromatic number; strong local colorable.

A strict strong coloring of trees

In this paper we introduce and study a new coloring parameter of a graph called the strict strong coloring (short SSColoring). A SSColoring of a graph G is a vertex proper coloring of G such that each vertex of G is adjacent to at least one not empty color class. The minimum number of colors among all SSColorings is called strict strong chromatic (short SSChromatic) number, denoted by χ s s ( G ) . In this paper we prove the NP-completeness of the problem, we discuss the χ s s ( G ) number of trees by giving some bounds. Finally, we give an optimal polynomial algorithm for SSColoring of trees.

Some results on [formula omitted]-choosability

A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph G is ( a : b ) -choosable, and c / d > a / b , then G is not necessarily ( c : d ) -choosable. Applying probabilistic methods, an upper bound for the k th choice number of a graph is given. We also prove that a directed graph with maximum outdegree d and no odd directed cycle is ( k ( d + 1 ) : k ) -choosable for every k ≥ 1 . Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability.

An improved bound for the strong chromatic number

AbstractLet η &gt; 0 be given. Then there exists d0 = d0(η) such that the following holds. Let G be a finite graph with maximum degree at most d ≥ d0 whose vertex set is partitioned into classes of size α d, where α≥ 11/4 + η. Then there exists a proper coloring of G with αd colors in which each class receives all αd colors. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:148–158, 2008

On the Strong Chromatic Number of Random Graphs

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding $k \big\lceil \frac{n}{k} \big\rceil - n$ isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

A Note on Adjacent Strong Edge Coloring of K(n,m)

In this paper, we prove that the adjacent strong edge chromatic number of a graph K(n,m) is n + 1, with n ≥ 2,m ≥ 1.

On the Strong Chromatic Number of Graphs

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph into disjoint subsets $V_1, \ldots, V_{\lceil n/k\rceil}$ of size $k$ each, one can find a proper $k$-vertex-coloring of the graph such that each part $V_i$, $i=1, \ldots, \lceil n/k\rceil$, contains exactly one vertex of each color. For any graph $G$ with maximum degree $\Delta$, it is easy to see that $\chi_S(G)\geq\Delta+1$. Recently, Haxell proved that $\chi_S(G) \leq 3\Delta -1$. In this paper, we improve this bound for graphs with large maximum degree. We show that $\chi_S(G)\leq 2\Delta$ if $\Delta \geq n/6$ and prove that this bound is sharp.