Chromatic Graphs Research Articles

On the Chromatic Number of $$\mathbb {R}^4$$ R 4

The lower bound for the chromatic number of $$\mathbb {R}^4$$ is improved from $$7$$ to $$9$$ . Three graphs with unit distance embeddings in $$\mathbb {R}^4$$ are described. The first is a $$7$$ -chromatic graph of order $$14$$ whose chromatic number can be verified by inspection. The second is an $$8$$ -chromatic graph of order $$26$$ . In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a $$9$$ -chromatic graph of order $$65$$ for which computer verification takes about one minute.

Chromatic numbers of graphs — large gaps

We say that a graph G is (?0,?)-chromatic if Chr(G) = ?, while Chr(G?) ≤ ?0 for any subgraph G? of G of size < |G|. The main result of this paper reads as follows. If ??+CH? holds for a given uncountable cardinal ?, then for every cardinal ?≤?, there exists an (?0,?)-chromatic graph of size ?+. We also study (?0,?+)-chromatic graphs of size ?+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal ?, there exists an (?0,?+)-chromatic graph of size ?+.

Extremal Bicyclic 3-Chromatic Graphs

A partial order relation in the set $$\mathcal {G}(n,k)$$G(n,k) of graphs of order $$n$$n and chromatic number $$k$$k can be defined as follows: Let $$G$$G and $$H$$H be two graphs in $$\mathcal {G}(n,k)$$G(n,k). $$G$$G is said to be less than $$H$$H if $$c_i(G)\le c_i(H)$$ci(G)≤ci(H) holds for every $$i$$i, $$k\le i\le n$$k≤i≤n and at least one inequality is strict, where $$c_i(G)$$ci(G) denotes the number of $$i$$i-color partitions of $$G$$G. These numbers are the coefficients of the chromatic polynomial in factorial form. In (J Graph Theory 43:210---222, 2003) the first $$\lceil n/2\rceil $$?n/2? levels of the diagram of the partially ordered set of connected 3-chromatic graphs of order $$n$$n were described. In this paper the previous work is continued and a description of the $$(\lceil n/2\rceil +1)$$(?n/2?+1)-st level is given; it contains $$n/2+1$$n/2+1 bicyclic graphs for even $$n$$n and $$(n-1)/2$$(n-1)/2 bicyclic graphs for odd $$n$$n. Some consequences concerning ordering chromatic polynomials of these graphs are deduced.

On the size of critical graphs with small maximum degree

In this paper, we give some new properties of edge chromatic critical graphs, and give new lower bounds for the average degree of Δ-critical graphs with Δ=11, 12 by the use of these properties.

Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone

We investigate the completely positive semidefinite cone ${\mathcal{CS}_{+}^n}$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular, we study relationships between this cone and the completely positive and the doubly nonnegative cone, and between its dual cone and trace positive noncommutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters $\alpha(G)$ and $\chi(G)$, which are roughly obtained by allowing variables to be positive semidefinite matrices instead of $0/1$ scalars in the programs defining the classical parameters. We can formulate these quantum parameters as conic linear programs over the cone ${\mathcal{CS}_{+}^n}$. Using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials.

Minors in ‐ Chromatic Graphs, II

AbstractLet denote the graph obtained from the complete graph by deleting the edges of some ‐subgraph. The author proved earlier that for each fixed s and , every graph with chromatic number has a minor. This confirmed a partial case of the corresponding conjecture by Woodall and Seymour. In this paper, we show that the statement holds already for much smaller t, namely, for .

A note on chromatic number and connectivity of infinite graphs

Consistently there exists an uncountably chromatic graph of cardinality ℵ1 which does not contain an uncountable infinitely connected subgraph.

A step towards the strong version of Havelʼs three color conjecture

In 1970, Havel asked if each planar graph with the minimum distance, d∇, between triangles large enough is 3-colorable. There are 4-chromatic planar graphs with d∇=3 (Aksenov, Melʼnikov, and Steinberg, 1980). The first result in the positive direction of Havelʼs problem was made in 2003 by Borodin and Raspaud, who proved that every planar graph with d∇⩾4 and no 5-cycles is 3-colorable.Recently, Havelʼs problem was solved by Dvořák, Králʼ and Thomas in the positive, which means that there exists a constant d such that each planar graph with d∇⩾d is 3-colorable. (As far as we can judge, this d is very large.)We conjecture that the strongest possible version of Havelʼs problem (SVHP) is true: every planar graph with d∇⩾4 is 3-colorable. In this paper we prove that each planar graph with d∇⩾4 and without 5-cycles adjacent to triangles is 3-colorable. The readers are invited to prove a stronger theorem: every planar graph with d∇⩾4 and without 4-cycles adjacent to triangles is 3-colorable, which could possibly open way to proving SVHP.

THE HOMOLOGY OF DIGRAPHS AS A GENERALIZATION OF HOCHSCHILD HOMOLOGY

Przytycki has established a connection between the Hochschild homology of an algebra A and the chromatic graph homology of a polygon graph with coefficients in A. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary A–A bimodule, for A possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.

A Generalization of Heterochromatic Graphs and f-Chromatic Spanning Forests

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.

Review of chromatic graph theory by Gary Chartrand and Ping Zhang

Review of chromatic graph theory by Gary Chartrand and Ping Zhang

The fractional version of Hedetniemi’s conjecture is true

This paper proves that the fractional version of Hedetniemi’s conjecture is true. Namely, for any graphs G and H , χ f ( G × H ) = min { χ f ( G ) , χ f ( H ) } . As a by-product, we obtain a proof of the Burr–Erdős–Lovász conjecture: For any positive integer n , there exists an n -chromatic graph G whose chromatic Ramsey number equals ( n − 1 ) 2 + 1 .

Complete and Almost Complete Minors in Double-Critical $8$-Chromatic Graphs

A connected $k$-chromatic graph $G$ is said to be double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A longstanding conjecture of Erdős and Lovász states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [Electron. J. Combin., 17(1): Research Paper 87, 2010] proved that every double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$ minor. It remains unknown whether an arbitrary double-critical $8$-chromatic graph contains a $K_8$ minor, but in this paper we prove that any double-critical $8$-chromatic contains a minor isomorphic to $K_8$ with at most one edge missing. In addition, we observe that any double-critical $8$-chromatic graph with minimum degree different from $10$ and $11$ contains a $K_8$ minor.

Ohba’s conjecture for graphs with independence number five

Ohba has conjectured that if G is a k -chromatic graph with at most 2 k + 1 vertices, then the list chromatic number or choosability ch ( G ) of G is equal to its chromatic number χ ( G ) , which is k . It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k -partite graph and each part has at most five vertices.

On colorability of graphs with forbidden minors along paths and circuits

Graphs distinguished by K r -minor prohibition limited to subgraphs induced by circuits have chromatic number bounded by a function f ( r ) ; precise bounds on f ( r ) are unknown. If minor prohibition is limited to subgraphs induced by simple paths instead of circuits, then for certain forbidden configurations, we reach tight estimates. A graph whose simple paths induce K 3 , 3 -minor free graphs is proven to be 6 -colorable; K 5 is such a graph. Consequently, a graph whose simple paths induce planar graphs is 6 -colorable. We suspect the latter to be 5 -colorable and we are not aware of such 5 -chromatic graphs. Alternatively, (and with more accuracy) a graph whose simple paths induce { K 5 , K 3 , 3 − } -minor free graphs is proven to be 4 -colorable (where K 3 , 3 − is the graph obtained from K 3 , 3 by removing a single edge); K 4 is such a graph.

Immersing small complete graphs

Following in the spirit of the Hadwiger and Hajós conjectures, Abu-Khzam and Langston have conjectured that every k -chromatic graph contains an immersion of K k . They proved this for k ≤ 4. Much before that, Lescure and Meyniel [F. Lescure and H. Meyniel, On a problem upon configurations contained in graphs with given chromatic number, Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), 325−331, Ann. Discrete Math. 41, North-Holland, Amsterdam, 1989] obtained a stronger result that included also the values k = 5 and 6, by proving that every simple graph of minimum degree k − 1 contains an immersion of K k . They noted that they also have a proof of the same result for k = 7 but have not published it due to the length of the proof. We give a simple proof of this result. This, in particular, proves the conjecture of Abu-Khzam and Langston for every k ≤ 7.

Double-Critical Graphs and Complete Minors

A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erdős and Lovász. The conjecture has been verified for $k$ at most $5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is $6$-connected and contains a complete $k$-graph as a minor.

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

AbstractWe investigate the pair-splitting numberwhich is a variation of splitting number, pair-reaping numberwhich is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants ofFσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

A conjecture on critical graphs and connections to the persistence of associated primes

We introduce a conjecture about constructing critically ( s + 1 ) -chromatic graphs from critically s -chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I in a polynomial ring R , i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass ( R / I s ) ⊆ Ass ( R / I s + 1 ) for all s ≥ 1 . To support our conjecture, we prove that the statement is true if we also assume that χ f ( G ) , the fractional chromatic number of the graph G , satisfies χ ( G ) − 1 < χ f ( G ) ≤ χ ( G ) . We give an algebraic proof of this result.

Dense graphs have [formula omitted] minors

Let K 3 , t ∗ denote the graph obtained from K 3 , t by adding all edges between the three vertices of degree t in it. We prove that for each t ≥ 6300 and n ≥ t + 3 , each n -vertex graph G with e ( G ) > 1 2 ( t + 3 ) ( n − 2 ) + 1 has a K 3 , t ∗ -minor. The bound is sharp in the sense that for every t , there are infinitely many graphs G with e ( G ) = 1 2 ( t + 3 ) ( | V ( G ) | − 2 ) + 1 that have no K 3 , t -minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every ( s + t ) -chromatic graph has a K s , t -minor.