Chromatic Graphs Research Articles

On the number of maximal independent sets in minimum colorings of split graphs

The largest number of color classes that are dominating sets (or, equivalently, maximal independent sets) among all χ(G)-colorings of a graph G is called the dominating-χ-color number of G, and denoted dχ(G). It was shown in Arumugam et al. (2011) that determining whether dχ(G)≥2 is NP-complete even in 3-chromatic graphs G. In this note, we prove that in split graphs the dominating-χ-color number equals to 1 if and only if its domination number is greater than 2. While such graphs can be efficiently recognized, we prove that for split graphs with domination number at most 2 the decision version of the problem of determining dominating-χ-color number is NP-complete. We also present existence results for split graphs attaining prescribed values of dominating-χ-color numbers and some other relevant parameters.

Patterns in Khovanov link and chromatic graph homology

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme–Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

Many cliques in $H$-free subgraphs of random graphs

For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $\chi(H)-1$ partite subgraph of $G(n,p)$. The transition between these two behaviors occurs at $p=n^{-1/m_2(H)}$. When $m_2(H) 0$ small at $p=n^{-1/m_2(H)+\delta}$ one can typically still keep most of the copies of $K_m$ in an $H$-free subgraph of $G(n,p)$. Thus, the transition between the two behaviors in this case occurs at some $p$ significantly bigger than $n^{-1/m_2(H)}$. To show that the second case is not redundant we present a construction which may be of independent interest. For each $k \geq 4$ we construct a family of $k$ chromatic graphs $G(k,\epsilon_i)$ where $m_2(G(k,\epsilon_i))$ tends to $\frac{(k+1)(k-2)}{2(k-1)} (< m_2(K_{k-1}))$ as $i$ tends to infinity. This is tight for all values of $k$

$K_5^-$-Subdivision in 4-Connected Graphs

Hajos conjectured in 1961 that every $k$-chromatic graph contains a $K_k$-subdivision. In this paper, we consider the subdivision of $K_5^-$ and prove that every 4-connected graph contains a $K_5^-...

A valency bound for distance-regular graphs

The regular complete t-partite graphs Kt×s (s,t positive integers at least 2) with valency k=(t−1)s have smallest eigenvalue −s=−k/(t−1), and hence, for fixed t there are infinitely many of them. In this paper we will show that these graphs are exceptional graphs for the class of distance-regular graphs. For this we will show a valency bound for distance-regular graphs with a relatively large, in absolute value, smallest eigenvalue. Using this bound, we classify the non-bipartite distance-regular graphs with diameter at most three with smallest eigenvalue not larger than −k/2, where k is the valency of the graph. As an application we complete the classification of the 3-chromatic distance-regular graphs with diameter three, which was started by Blokhuis, Brouwer and Haemers.

Maximizing the number of [formula omitted]-colorings of [formula omitted]-chromatic graphs

Let C4(n) be the family of all connected 4-chromatic graphs of order n. Given an integer x≥4, we consider the problem of finding the maximum number of x-colorings of a graph in C4(n). It was conjectured that the maximum number of x-colorings is equal to (x)↓4(x−1)n−4 and the extremal graphs are those which have clique number 4 and size n+2.In this article, we reduce this problem to a finite family of graphs. We show that there exists a finite family F of connected 4-chromatic graphs such that if the number of x-colorings of every graph G in F is less than (x)↓4(x−1)|V(G)|−4 then the conjecture holds to be true.

Optimal Colorings with Rainbow Paths

Let G be a connected graph of chromatic number k. For a k-coloring f of G, a full f-rainbow path is a path of order k in G whose vertices are all colored differently by f. We show that G has a k-coloring f such that every vertex of G lies on a full f-rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper vertex-colorings, Graphs Combin. 23 2007, 201–203). Furthermore, we show that if G has a cycle of length 0 modulo k, then G has a k-coloring f such that, for every vertex u of G, some full f-rainbow path begins at u, which solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic graphs, arXiv:1503.00965v1 ) and verifies some special cases of a conjecture of Akbari et al. (Colorful paths in vertex coloring of graphs, preprint). Finally, we establish some more results on the existence of optimal colorings with (directed) full rainbow paths.

On ordinary and weighted Harary indices of Mycielski graphs

The Mycielski’s construction is a well-known construction on graphs which transforms a [Formula: see text]-chromatic triangle-free graph [Formula: see text] into a [Formula: see text]-chromatic triangle-free graph [Formula: see text] which is called the Mycielski graph of [Formula: see text]. In this paper, we compute the Harary index, additively weighted Harary index, and multiplicatively weighted Harary index of the Mycielski graph of any simple connected graph with girth greater than [Formula: see text].

On the size of edge chromatic 5-critical graphs

In this paper, we study the size of edge chromatic 5-critical graphs in several classes of 5-critical graphs. In most of the classes of 5-critical graphs in this paper, we have obtained their exact size and in the other classes of 5-critical graphs, we give new bounds on their number of major vertices and size.

Colorful paths for 3-chromatic graphs

In this paper, we prove that every 3-chromatic connected graph, except C7, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path with 3 vertices. It is a special case of a conjecture due to S. Akbari, F. Khaghanpoor, and S. Moazzeni stating that every connected graph G other than C7 admits a χ(G)-coloring such that every vertex of G is the beginning of a colorful path (i.e. a path on χ(G) vertices containing a vertex of each color). We also provide some support for the conjecture in the case of 4-chromatic graphs.

Hesitancy Fuzzy Graph Coloring

Graph coloring dates back to 1852, when Francis Guthrie come up with the four color conjecture. Gary Chartrand and Ping Zhang [3] discussed various colorings of graph and its properties in their book entitled Chromatic Graph Theory. A graph coloring is the assignment of a color to each of the vertices or edges or both in such a way that no two adjacent vertices and incident edges share the same color. Graph coloring has been applied to many real world problems like scheduling, allocation, telecommunications and bioinformatics, etc.

Critical injective chromatic graphs

The concept of critical injective chromatic graph is introduced. Some properties of critical injective chromatic graph are studied based on number of vertices and degree of graphs. The condition for cycles to be critical injective chromatic is obtained. The existence of a critical injective chromatic graph is established. An algorithm for injective colouring of vertices of a given connected graph is proposed. An application of injective chromatic number and injective chromatic sum in the diagnosis of diseases based on symptoms is suggested.

Induced Colorful Trees and Paths in Large Chromatic Graphs

In a proper vertex coloring of a graph a subgraph is colorful if its vertices are colored with different colors. It is well-known (see for example in Gyárfás (1980)) that in every proper coloring of a $k$-chromatic graph there is a colorful path $P_k$ on $k$ vertices. The first author proved in 1987 that $k$-chromatic and triangle-free graphs have a path $P_k$ which is an induced subgraph. N.R. Aravind conjectured that these results can be put together: in every proper coloring of a $k$-chromatic triangle-free graph, there is an induced colorful $P_k$. Here we prove the following weaker result providing some evidence towards this conjecture: For a suitable function $f(k)$, in any proper coloring of an $f(k)$-chromatic graph of girth at least five, there is an induced colorful path on $k$ vertices.

Weighted graphs: Eigenvalues and chromatic number

We revisit Hoffman relation involving chromatic number $\chi$ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues $\lambda$ dan $\mu$ satisfy $\lambda=(1-\chi)\mu.$ We study in particular the eigenvalues of the integer simplex $T_m^2,$ a 3-chromatic graph on $\binom {m+2}{2}$ vertices.

Hadwiger’s Conjecture for inflations of 3-chromatic graphs

Hadwiger’s Conjecture states that every k-chromatic graph has a complete minor of order k. A graph G′ is an inflation of a graph G if G′ is obtained from G by replacing each vertex v of G by a clique Cv and joining two vertices of distinct cliques by an edge if and only if the corresponding vertices of G are adjacent. We present an algorithm for computing an upper bound on the chromatic number χ(G′) of any inflation G′ of any 3-chromatic graph G. As a consequence, we deduce that Hadwiger’s Conjecture holds for any inflation of any 3-colorable graph.

Trees, ladders and graphs

We introduce a new method to construct uncountably chromatic graphs from non-special trees and ladder systems. Answering a question of P. Erdős and A. Hajnal from 1985, we construct graphs of chromatic number ω1 without uncountable ω-connected subgraphs. Second, we build triangle free graphs of chromatic number ω1 without subgraphs isomorphic to Hω,ω+2.

Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth

In a recent work (Kaiser et al., J. Comb. Theory Ser. A 123, 239–251, 2014), Kaiser et al. provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs and, thus, give counterexamples to a conjecture of Francisco et al. (Discrete Math. 310, 2176–2182, 2010). The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.

An Extension of the Blow-up Lemma to Arrangeable Graphs

The blow-up lemma established by Komlos, Sarkozy, and Szemeredi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalizations of this result concerning the embedding of $a$-arrangeable graphs, where a graph is called $a$-arrangeable if its vertices can be ordered in such a way that the neighbors to the right of any vertex $v$ have at most $a$ neighbors to the left of $v$ in total. Examples of arrangeable graphs include planar graphs and, more generally, graphs without a $K_s$-subdivision for constant $s$. Our main result shows that $a$-arrangeable graphs with maximum degree at most $\sqrt{n}/\log n$ can be embedded into corresponding systems of superregular pairs. This is optimal up to the logarithmic factor. We also present two applications. We prove that any large enough graph $G$ with minimum degree at least $\big(\frac{r-1}{r}+\gamma\big)n$ contains an $F$-factor of every $a$-arrangeable $r$-chromatic graph $F$ with at most $\xi n...

Chromatic kernel and its applications

In this paper, we study the following Chromatic kernel (CK) problem: given an $$n$$n-partite graph (called a chromatic correlation graph) $$G=(V,E)$$G=(V,E) with $$V=V_{1}\bigcup \cdots \bigcup V_{n}$$V=V1???Vn and each partite set $$V_{i}$$Vi containing a constant number $$\lambda $$? of vertices, compute a subgraph $$G[V_{CK}]$$G[VCK] of $$G$$G with exactly one vertex from each partite set and the maximum number of edges or the maximum total edge weight, if $$G$$G is edge-weighted (among all such subgraphs). CK is a new problem motivated by several applications and no existing algorithm directly solves it. In this paper, we first show that CK is NP-hard even if $$\lambda =2$$?=2, and cannot be approximated within a factor of $$16/17$$16/17 unless P = NP. Then, we present a random-sampling-based PTAS for dense CK. As its application, we show that CK can be used to determine the pattern of chromosome associations in the nucleus for a population of cells. We test our approach by using random and real biological data; experimental results suggest that our approach yields near optimal solutions, and significantly outperforms existing approaches.

Adjacency preservers on invertible hermitian matrices I

Fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction.In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and finite geometry.