Chromatic Graphs Research Articles

On the Biplanarity of Blowups

The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar: iterated Kleetopes are counterexamples. Additionally, we construct biplanar drawings of 2-blowups of planar graphs whose duals have two-path induced path partitions, and drawings with split thickness two of 2-blowups of 3-chromatic planar graphs, and of graphs that can be decomposed into a Hamiltonian path and a dual Hamiltonian path.

4-Chromatic Graphs Have At Least Four Cycles of Length $0 \bmod 3$

4-Chromatic Graphs Have At Least Four Cycles of Length $0 \bmod 3$

Hajnal–Máté graphs, Cohen reals, and disjoint‐type guessing

AbstractA Hajnal–Máté graph is an uncountably chromatic graph on satisfying a certain natural sparseness condition. We investigate Hajnal–Máté graphs and generalizations thereof, focusing on the existence of Hajnal–Máté graphs in models resulting from adding a single Cohen real. In particular, answering a question of Dániel Soukup, we show that such models necessarily contain triangle‐free Hajnal–Máté graphs. In the process, we isolate a weakening of club guessing called disjoint‐type guessing that we feel is of interest in its own right. We show that disjoint‐type guessing is independent of and, if disjoint‐type guessing holds in the ground model, then the forcing extension by a single Cohen real contains Hajnal–Máté graphs such that the chromatic numbers of finite subgraphs of grow arbitrarily slowly.

Edge geo chromatic number of a graph

The Edge Geodetic Number is also known as the smallest size among all sets of edges that make up Edge Geodetic sets and is represented by the symbol g1(G). Assuming G is a finite connected graph, we can define a set S contained in V(G) as an Edge Geodetic set if, for any pair of vertices u and v in set S, every edge in G lies on the shortest path between u and v. A set C contained in E(G) is referred to as a chromatic index set when it contains all of the proper k-edge colors of G. The chromatic index is determined by the set's lowest cardinality among its edges. The "Chromatic Index Number" is a term used to refer to the collection of these minimum cardinality chromatic index values and is represented by the symbol chi'(G). By combining an Edge Geodetic set with a Chromatic Index set, the Edge Geo Chromatic set of G is created, providing an innovative concept. The Edge Geo Chromatic Number, abbreviated as X, is the least cardinality among all Edge Geo Chromatic sets of G. In essence, an Edge Geodetic set and a Chromatic Index set combine to form an Edge Geo Chromatic graph. We talk about an edge geo chromatic linked graph with n orders. For a number of common graphs, we have computed the Edge Geo Chromatic Number and established its bounds. Furthermore, we have shown that when G represents a graph of diameter d, then chi'_gc(G) lessthan or equal to n - d +2.

Counterexamples to Gerbner's conjecture on stability of maximal F‐free graphs

AbstractLet and let be an ‐chromatic graph with a color‐critical edge, that is, there exists such that has chromatic number . Gerbner recently conjectured that every ‐vertex maximal ‐free graph with at least edges contains an induced complete ‐partite graph on vertices. Let be a graph obtained from copies of by sharing a common edge. In this paper, we show that for all if is an ‐vertex maximal ‐free graph with at least edges, then contains an induced complete bipartite graph on vertices. We also construct graphs to show that the magnitude in is best possible, which disproves Gerbner's conjecture for .

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.

Bad list assignments for non‐k $k$‐choosable k $k$‐chromatic graphs with 2k+2 $2k+2$‐vertices

AbstractIt was conjectured by Ohba, and proved by Noel, Reed and Wu that ‐chromatic graphs with are chromatic‐choosable. This upper bound on is tight: if is even, then and are ‐chromatic graphs with vertices that are not chromatic‐choosable. It was proved by Zhu and Zhu that these are the only non‐‐choosable complete ‐partite graphs with vertices. For , a bad list assignment of is a ‐list assignment of such that is not ‐colourable. Bad list assignments for were characterized by Enomoto, Ohba, Ota and Sakamoto. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for . Using these results, we characterize all non‐‐choosable ‐chromatic graphs with vertices.

Generalized Turán problems for double stars

We study the generalized Turán function ex(n,H,F), when H or F is a double star Sa,b, which is a tree with a central edge uv, a leaves connected to u and b leaves connected to v. We determine ex(n,Kk,Sa,b) and ex(n,Sa,b,F) for sufficiently large n, where F is either a 3-chromatic graph with an edge whose deletion results in a bipartite graph, or the 2-fan, i.e. two triangles sharing a vertex. We also give bounds on ex(n,Sa,b,Sc,d).

Triangles in graphs without bipartite suspensions

Given graphs T and H, the generalized Turán number ex(n,T,H) is the maximum number of copies of T in an n-vertex graph with no copies of H. Alon and Shikhelman, using a result of Erdős, determined the asymptotics of ex(n,K3,H) when the chromatic number of H is greater than three and proved several results when H is bipartite. We consider this problem when H has chromatic number three. Even this special case for the following relatively simple three chromatic graphs appears to be challenging. The suspension Hˆ of a graph H is the graph obtained from H by adding a new vertex adjacent to all vertices of H. We give new upper and lower bounds on ex(n,K3,Hˆ) when H is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.

Minimum Degree Stability of H-Free Graphs

Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 - 1/r + o(1)) binom{n}{2} edges, and any H-free graph with that many edges can be made r-partite by deleting o(n^{2}) edges. Here we consider a natural variant of this—the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n^{2}) edges? We determine this least value for all 3-chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the Andrásfai-Erdős-Sós theorem and work of Alon and Sudakov.

Bipolar Triangular Neutrosophic Chromatic Numbers with the Application of traffic light system

For addressing issues in several domains, such as theoretical computer science, engineering, physics, combinatorics, and the medical sciences, graph theory is a crucial component of mathematics. Graph coloring is one of the new settings that is emerging in a neutrosophic chromatic number environment. In addition to introducing the idea of bipolar triangular neutrosophic chromatic graphs (BTNCG), this work also examines and demonstrates the algebraic assumption. The proposed concept has been applied in a traffic signal system to discover a new lane to avoid traffic in peak hours.

Quasi-Shor Algorithms for Global Benchmarking of Universal Quantum Processors

This work generalizes Shor’s algorithm into quasi-Shor algorithms by replacing the modular exponentiation with alternative unitary operations. By using the quantum circuits to generate Bell states as the unitary operations, a specific example called the Bell–Shor algorithm was constructed. The system density matrices in the quantum circuits with four distinct input states were calculated in ideal conditions and illustrated through chromatic graphs to witness the evolution of quantum states in the quantum circuits. For the real part of the density matrices, it was revealed that the number of zero elements dramatically declined to only a few points after the operation of the inverse quantum Fourier transformation. Based on this property, a protocol constituting a pair of error metrics Γa and Γb is proposed for the global benchmarking of universal quantum processors by looking at the locations of the zero entries and normalized average values of non-zero entries. The protocol has polynomial resource requirements with the scale of the quantum processor. The Bell–Shor algorithm is capable of being a feasible setting for the global benchmarking of universal quantum processors.

Extremal Khovanov homology and the girth of a knot

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text]

Pumping schedule assignment using chromatic graphs to reduce groundwater pumping energy consumption

Abstract In this paper, groundwater pumping schedules are examined, in terms of pumping energy consumption minimization. Chromatic graphs are used for the first time to assign groundwater pumping schedules. Assuming that a finite number of users pumps from a common aquifer, we examine the following three pumping scenarios: (a) all users pump simultaneously, (b) users pump during one out of two different periods, (c) users pump during one out four different periods. A computational code, based on Theis equation, is constructed to calculate the drawdowns and, subsequently, the energy consumption, for various well locations, numbers of wells, flow rates, pumping period durations and aquifers' characteristics. We use nearest neighbor graphs and chromatic graphs to represent the interactions of the well users. We compare the results obtained when chromatic graphs are used, to the optimum, which is obtained using genetic algorithms (GAs). Our results indicate that: (a) performance of chromatic graphs is good, (b) when two pumping periods are used, it is sufficient for each player to cooperate with their nearest neighbor and pump alternately. In typical cases, the energy consumption reduction can be around 10–40%. The benefit is even higher if four pumping periods are used.

Some Remarks on Even-Hole-Free Graphs

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erdős-Lovász Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that every even-hole-free graph $G$ with $\omega(G)<\chi(G)=s+t-1$ satisfies the Erdős-Lovász Tihany conjecture provided that $ t\geq s > \chi(G)/3$; every $9$-chromatic graph $G$ with $\omega(G)\leq 8$ has a $K_4\cup K_6$ minor; for all $k\geq 7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs.

On Chromatic Vertex Stability of 3-Chromatic Graphs With Maximum Degree 4

The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $\chi(G-S)=\chi(G)-1$. In this paper we construct infinitely many graphs $G$ with $\Delta(G)=4$, $\chi(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.

Every Orientation of a $4$-Chromatic Graph has a Non-Bipartite Acyclic Subgraph

Let $f(n)$ denote the smallest integer such that every directed graph with chromatic number larger than $f(n)$ contains an acyclic subgraph with chromatic number larger than $n$. The problem of bounding this function was introduced by Addario-Berry et al., who noted that $f(n) \leqslant n^2$. The only improvement over this bound was obtained by Nassar and Yuster, who proved that $f(2)=3$ using a deep theorem of Thomassen. Yuster asked if this result can be proved using elementary methods. In this short note we provide such a proof.

Local boxicity

A box is the cartesian product of real intervals, which are either bounded or equal to R. A box is said to be d-local if at most d of the intervals are bounded. In this paper, we investigate the recently introduced local boxicity of a graph G, which is the minimum d such that G can be represented as the intersection of d-local boxes in some dimension. We prove that all graphs of maximum degree Δ have local boxicity O(Δ), while almost all graphs of maximum degree Δ have local boxicity Ω(Δ), improving known upper and lower bounds. We also give improved bounds on the local boxicity as a function of the number of edges or the genus. Finally, we investigate local boxicity through the lens of chromatic graph theory. We prove that the family of graphs of local boxicity at most 2 is χ-bounded, which means that the chromatic number of the graphs in this class can be bounded by a function of their clique number. This extends a classical result on graphs of boxicity at most 2.

Dichromatic number and forced subdivisions

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F, denote by maderχ→(F) the smallest integer k such that every k-dichromatic digraph contains a subdivision of F. As our first main result, we prove that if F is an orientation of a cycle then maderχ→(F)=v(F). This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests.Our second main result is that maderχ→(F)=4 for every tournament F of order 4. This is an extension of the classical result by Dirac that 4-chromatic graphs contain a K4-subdivision to directed graphs.

Further evidence towards the multiplicative 1-2-3 Conjecture

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings.In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vučković can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes.