Chromatic Equivalence Research Articles

Statistical Properties of Color Matching Functions.

In trichromats, color vision entails the projection of an infinite-dimensional space (the one containing all possible electromagnetic power spectra) onto the three-dimensional space that modulates the activity of the three types of cones. This drastic reduction in dimensionality gives rise to metamerism, that is, the perceptual chromatic equivalence between two different light spectra. The classes of equivalence of metamerism are revealed by color-matching experiments in which observers adjust the intensity of three monochromatic light beams of three preset wavelengths (the primaries) to produce a mixture that is perceptually equal to a given monochromatic target stimulus. Here we use the linear relation between the color matching functions and the absorption probabilities of each type of cone to find particularly useful triplets of primaries. As a second goal, we also derive an analytical description of the trial-to-trial variability and the correlations of color matching functions stemming from Poissonian noise in photon capture. We analyze how the statistical properties of the responses to color-matching experiments vary with the retinal composition and the wavelengths of peak absorption probability, and compare them with experimental data on subject-to-subject variability obtained previously.

Short certificates for chromatic equivalence

The chromatic polynomial gives the number of proper colourings of a graph in terms of the number of available colours. In general, calculating chromatic polynomials is #P-hard. Two graphs are chromatically equivalent if they have the same chromatic polynomial. At present, determining if two graphs are chromatically equivalent involves computation and comparison of their chromatic polynomials, or similar computational effort. In this paper we investigate a new approach, certificates of chromatic equivalence, first proposed by Morgan and Farr. These give proofs of chromatic equivalence, without directly computing the polynomials. The lengths of these proofs may provide insight into the computational complexity of chromatic equivalence and related problems including chromatic factorisation and chromatic uniqueness. For example, if the lengths of shortest certificates of chromatic equivalence are bounded above by a polynomial in the size of the graphs, then chromatic equivalence belongs to NP. After establishing some links of this type between certificate length and computational complexity, we give some theoretical and computational results on certificate length. We prove that, if the number of different chromatic polynomials falls well short of the number of different graphs, then for all sufficiently large $n$ there are pairs of chromatically equivalent graphs on $n$ vertices with certificate of chromatic equivalence of length $\Omega(n^2/\log n)$. We give a linear upper bound on shortest certificate length for trees. We designed and implemented a program for finding short certificates of equivalence using a minimal set of certificate steps. This program was used to find the shortest certificates of equivalence for all pairs of chromatically equivalent graphs of order $n\leq 7$.

The chromatic equivalence class of [formula omitted

The chromatic equivalence class of a graph G is the set of graphs that have the same chromatic polynomial as G. We find the chromatic equivalence class of the complete tripartite graphs K1,n,n+2 for all n≥2. This partially answers a question raised in Chia and Ho (2009), which asks for the chromatic equivalence class of the graph K1,m,n where 2≤m≤n.

Chromaticity of Turan Graphs with At Most Three Edges Deleted

Let P(G,�) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,�) = P(G,�) implies H is isomorphic to G. In this paper, we determine the chro- maticity of all Turan graphs with at most three edges deleted. As a by product, we found many families of chromatically unique graphs and chromatic equivalence classes of graphs.

CHROMATIC EQUIVALENCE OF A FAMILY OF K4-HOMEOMORPHS WITH GIRTH 9

For a graph G, let P (G,λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χ−equivalent), denoted by G ∼ H, if P (G,λ) = P (H,λ). A graph G is chromatically unique (or simply χ−unique) if for any graph H such as H ∼ G, we have H ∼= G, i.e, H is isomorphic to G. A K4-homeomorph is a subdivision of the complete graph K4. In this paper, we determine when two K4-homeomorphs of the form K4(2, 3, 4, d, e, f) and K4(1, 2, 6, d ′ , e, f ) are chromatically equivalent. The result obtained can be extended in the study of chromatic equivalence classes of K4(2, 3, 4, d, e, f) and chromatic uniqueness of K4-homeomorphs with girth 9. AMS Subject Classification: 05C15

Chromatically equivalent pairs of K 4-homeomorphic graphs

We determine when two K4-homeomorphs of the form K4(1, 2, c, d, e, f) are chromatically equivalent. The result obtained can be extended in the study of chromatic equivalence classes of K4(1, 2, c, d, e, f) and chromatic uniqueness of K4-homeomorphs with girth c + 3.

On chromatic equivalence of a pair of K4-homeomorphs

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent, denoted \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). We write \([G] = \{H| H \sim G\}\). If \([G] = \{G\}\), then \(G\) is said to be chromatically unique. In this paper, we discuss a chromatically equivalent pair of graphs in one family of \(K_4\)-homeomorphs, \(K_4(1, 2, 8, d, e, f)\). The obtained result can be extended in the study of chromatic equivalence classes of \(K_4(1, 2, 8, d, e, f)\) and chromatic uniqueness of \(K_4\)-homeomorphs with girth \(11\).

Chromatic classes of 2-connected [formula omitted]-graphs with three triangles and one induced 4-cycle

For a graph G , let P ( G , λ ) be its chromatic polynomial. Two graphs G and H are chromatically equivalent, denoted G ∼ H , if P ( G , λ ) = P ( H , λ ) . A graph G is chromatically unique if P ( H , λ ) = P ( G , λ ) implies that H ≅ G . In this paper, we shall determine all chromatic equivalence classes of 2-connected ( n , n + 4 ) -graphs with three triangles and one induced 4-cycle, under the equivalence relation ‘ ∼’. As a by product of these, we obtain various new families of chromatically-equivalent graphs and chromatically-unique graphs.

Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph K m , n , r are developed. Using these, we obtain the chromatic equivalence classes for K m , n , n (where 1 ≤ m ≤ n ) and K m 1 , m 2 , m 3 (where | m i − m j | ≤ 3 ). In particular, it is shown that (i) K m , n , n (where 2 ≤ m ≤ n ) and (ii) K m 1 , m 2 , m 3 (where | m i − m j | ≤ 3 , 2 ≤ m i , i = 1 , 2 , 3 ) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175–179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27–37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs K n 1 , n 2 , n 3 J. Systems Sci. Math. Sci. 20 (2000) 181–186].

The chromatic equivalence class of graph Bn-6,12

The chromatic equivalence class of graph B_n-6,12

The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than [formula omitted

Let G be a graph with order n and G ¯ its complement. Denote by β ( G ) the minimum real root of the adjoint polynomial of G . Two graphs G and H are chromatically equivalent if and only if G ¯ and H ¯ are adjointly equivalent. G is chromatically unique if and only if G ¯ is adjointly unique. In this paper, we give a method to determine all chromatic equivalence classes of a graph G with β ( G ¯ ) > - 4 , by using some results on the minimum real roots of the adjoint polynomial of G ¯ . Moreover, we obtain a necessary and sufficient condition for those graphs that are chromatically unique.

A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs

Let P ( G , λ ) be the chromatic polynomial of a graph G . A graph G is chromatically unique if for any graph H , P ( H , λ ) = P ( G , λ ) implies H ≅ G . Koh, Teo and Chia conjectured that for any integers n and k with n ⩾ k + 2 ⩾ 4 , the complete tripartite graph K ( n - k , n , n ) is chromatically unique. Let K ( n , m , r ) - S denote the graph obtained by deleting all edges in S from the complete tripartite K ( n , m , r ) . In this paper, we establish that for any positive integer n ⩾ m ⩾ r ⩾ 2 , the chromatic equivalence class of K ( n , m , r ) is contained in the family { K ( x , y , z ) - S | 1 ⩽ x ⩽ y ⩽ z , m ⩽ z ⩽ n , x + y + z = n + m + r , S ⊂ E ( K ( x , y , z ) ) and | S | = xy + xz + yz - nm - nr - mr }. By applying these results, we confirm this conjecture and show that K ( n - k , n - 1 , n ) is chromatically unique if n ⩾ 2 k and k ⩾ 2 .

Chromatic equivalence classes of certain generalized polygon trees, III

Let P( G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, if P( G)= P( H). A set of graphs S is called a chromatic equivalence class if for any graph H that is chromatically equivalent with a graph G in S , then H∈ S . Peng et al. (Discrete Math. 172 (1997) 103–114), studied the chromatic equivalence classes of certain generalized polygon trees. In this paper, we continue that study and present a solution to Problem 2 in Koh and Teo (Discrete Math. 172 (1997) 59–78).

Chromaticity of some families of dense graphs

For a graph G, let P( G, λ) be its chromatic polynomial and let [ G] be the set of graphs having P( G, λ) as their chromatic polynomial. We call [ G] the chromatic equivalence class of G. If [ G]={ G}, then G is said to be chromatically unique. In this paper, we first determine [ G] for each graph G whose complement G ̄ is of the form aK 1∪bK 3∪⋃ 1⩽i⩽s P l i , where a, b are any nonnegative integers and l i is even. By this result, we find that such a graph G is chromatically unique iff ab=0 and l i ≠4 for all i. This settles the conjecture that the complement of P n is chromatically unique for each even n with n≠4. We also determine [ H] for each graph H whose complement H ̄ is of the form aK 3∪⋃ 1⩽i⩽s P u i ∪ ⋃ 1⩽j⩽t C v j , where u i ⩾3 and u i≢4 ( mod 5) for all i. We prove that such a graph H is chromatically unique if u i +1≠ v j for all i, j and u i is even when u i ⩾6.

On the chromaticity of K4 homeomorphs

Let K 4( a, b, c, d, e, f) denote the K 4 homeomorph in which the lengths of its six paths between pairs of degree-3 vertices are a, b, c, d, e, f, respectively. In this paper, we study the chromatic uniqueness and chromatic equivalence of K 4( a, b, c, d, e, f) when exactly three of a, b, c, d, e, f are equal to s(⩾2).

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k-independent partitions of G. Let 𝒦−s(p,q) (resp., 𝒦2−s(p,q)) denote the family of connected (resp., 2-connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ 𝒦 −s(p,q) and 0 ≤ s ≤ (p − 1)(q − 1) (resp., G ∈ 𝒦 2−s(p,q) and 0 ≤ s ≤ p + q − 4). These bounds are then used to show that if G ∈ 𝒦 −s(p,q) (resp., G ∈ 𝒦 2−s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets 𝒦−si(p+i,q−i) where max and si = s − i(p−q+i) (resp., a subset of 𝒦2−s(p,q), where either 0 ≤ s ≤ q − 1, or s ≤ 2q − 3 and p ≥ q + 4). By applying these results, we show finally that any 2-connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q − 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

AbstractGiven a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let 𝒦−s(p,q) (resp., 𝒦2−s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ 𝒦 −s(p,q) and 0 ≤ s ≤ (p − 1)(q − 1) (resp., G ∈ 𝒦 2−s(p,q) and 0 ≤ s ≤ p + q − 4). These bounds are then used to show that if G ∈ 𝒦 −s(p,q) (resp., G ∈ 𝒦 2−s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets 𝒦−si(p+i,q−i) where max $\{-{s\over q-1},-{p-q\over 2}\}\!\le \,i \le{s\over p-1}$ and si = s − i(p−q+i) (resp., a subset of 𝒦2−s(p,q), where either 0 ≤ s ≤ q − 1, or s ≤ 2q − 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q − 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001

Chromatic equivalence classes of certain cycles with edges

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G∼H, if P(G)=P(H). A graph G is chromatically unique if for any graph H, G∼H implies that G is isomorphic with H. In this paper, we give the necessary and sufficient conditions for a family of generalized polygon trees to be chromatically unique.

An attempt to classify bipartite graphs by chromatic polynomials

For integers p, q, s with p⩾ q⩾3 and 1⩽ s⩽ q−1, let K −s(p,q) (resp. K 2 −s(p,q) ) denote the set of connected (resp. 2-connected) bipartite graphs which can be obtained from K p, q by deleting a set of s edges. In this paper, we first find an upper bound for the 3-independent partition number of a graph G∈ K −s(p,q) with respect to the maximum degree Δ( G′) of G′, where G′= K p, q − G. By using this result, we show that the set {G | G∈ K 2 −s(p,q),Δ(G′)=i} is closed under the chromatic equivalence for every integer i with s⩾ i⩾( s+3)/2. From this result, we prove that for any G∈ K 2 −s(p,q) with p⩾ q⩾3, if 5⩽ s⩽ q−1 and Δ( G′)= s−1, or 7⩽ s⩽ q−1 and Δ( G′)= s−2, then G is chromatically unique.

On the chromatic equivalence class of graphs

In this paper, the set of all graphs having the same chromatic polynomial as that of the graph consisting of the join of m copies of the paths each of length 3 is determined. Also, it is shown that, for i = 0, 1, the set of all graphs having the same chromatic polynomial ( λ - i) P( G; λ) is obtained by applying the two operations, described in the paper, to every graph which has the same chromatic polynomial as that of G. This result is used in the final section to prove a characterization for T-chromatically unique graphs.