scholarly journals The chromatic spectrum of 3-uniform bi-hypergraphs

2011 ◽  
Vol 311 (23-24) ◽  
pp. 2650-2656 ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang
Keyword(s):  
Chromatic Spectrum

scholarly journals Integumentary Colour Allocation in the Stork Family (Ciconiidae) Reveals Short-Range Visual Cues for Species Recognition

Birds ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 138-146
Author(s):  
Eduardo J. Rodríguez-Rodríguez ◽  
Juan J. Negro
Keyword(s):  
Short Range ◽  
Visual Cues ◽  
Species Recognition ◽  
The Body ◽  
Whole Body ◽  
Divergent Evolution ◽  
Black And White ◽  
Extant Species ◽  
Chromatic Spectrum ◽  
Mixed Species Flocks

The family Ciconiidae comprises 19 extant species which are highly social when nesting and foraging. All species share similar morphotypes, with long necks, a bill, and legs, and are mostly coloured in the achromatic spectrum (white, black, black, and white, or shades of grey). Storks may have, however, brightly coloured integumentary areas in, for instance, the bill, legs, or the eyes. These chromatic patches are small in surface compared with the whole body. We have analyzed the conservatism degree of colouration in 10 body areas along an all-species stork phylogeny derived from BirdTRee using Geiger models. We obtained low conservatism in frontal areas (head and neck), contrasting with a high conservatism in the rest of the body. The frontal areas tend to concentrate the chromatic spectrum whereas the rear areas, much larger in surface, are basically achromatic. These results lead us to suggest that the divergent evolution of the colouration of frontal areas is related to species recognition through visual cue assessment in the short-range, when storks form mixed-species flocks in foraging or resting areas.

On the chromatic spectrum of acyclic decompositions of graphs

2007 ◽  
Vol 56 (2) ◽  
pp. 83-104 ◽  
Author(s):  
Robert E. Jamison ◽  
Eric Mendelsohn
Keyword(s):  
Chromatic Spectrum

scholarly journals The Smallest One-Realization of a Given Set

10.37236/1171 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang
Keyword(s):  
Open Problem ◽  
Feasible Set ◽  
Minimum Number ◽  
Chromatic Spectrum ◽  
Positive Integers ◽  
Mixed Hypergraph

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we  determine the number  of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král  in 2004.

scholarly journals On the Upper and Lower Chromatic Numbers of BSQSs(16)

10.37236/1550 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Giovanni Lo Faro ◽  
Lorenzo Milazzo ◽  
Antoinette Tripodi
Keyword(s):  
Chromatic Number ◽  
Chromatic Numbers ◽  
New Information ◽  
Chromatic Spectrum ◽  
Minimum Numbers ◽  
Mixed Hypergraph

A mixed hypergraph is characterized by the fact that it possesses ${\cal C}$-edges as well as ${\cal D}$-edges. In a colouring of a mixed hypergraph, every ${\cal C}$-edge has at least two vertices of the same colour and every ${\cal D}$-edge has at least two vertices coloured differently. The upper and lower chromatic numbers $\bar{\chi}$, $\chi$ are the maximum and minimum numbers of colours for which there exists a colouring using all the colours. The concepts of mixed hypergraph, upper and lower chromatic numbers are applied to $SQSs$. In fact a BSQS is an SQS where all the blocks are at the same time ${\cal C}$-edges and ${\cal D}$-edges. In this paper we prove that any $BSQS(16)$ is colourable with the upper chromatic number $\bar{\chi}=3$ and we give new information about the chromatic spectrum of BSQSs($16$).

The minimum chromatic spectrum of 3-uniform $$\mathcal{C}$$ C -hypergraphs

2013 ◽  
Vol 29 (4) ◽  
pp. 796-802
Author(s):  
Ruixue Zhang ◽  
Ping Zhao ◽  
Kefeng Diao ◽  
Fuliang Lu
Keyword(s):  
Chromatic Spectrum

Chromatic spectrum of some classes of 2-regular bipartite colored graphs

2021 ◽  
pp. 1-8
Author(s):  
Muhammad Imran ◽  
Yasir Ali ◽  
Mehar Ali Malik ◽  
Kiran Hasnat
Keyword(s):  
Adjacency Matrix ◽  
Colored Graph ◽  
Colored Graphs ◽  
Chromatic Spectrum ◽  
General Graphs ◽  
Disconnected Graph

Chromatic spectrum of a colored graph G is a multiset of eigenvalues of colored adjacency matrix of G. The nullity of a disconnected graph is equal to sum of nullities of its components but we show that this result does not hold for colored graphs. In this paper, we investigate the chromatic spectrum of three different classes of 2-regular bipartite colored graphs. In these classes of graphs, it is proved that the nullity of G is not sum of nullities of components of G. We also highlight some important properties and conjectures to extend this problem to general graphs.

scholarly journals The chromatic spectrum of signed graphs

2016 ◽  
Vol 339 (11) ◽  
pp. 2660-2663 ◽  
Author(s):  
Yingli Kang ◽  
Eckhard Steffen
Keyword(s):  
Signed Graphs ◽  
Chromatic Spectrum

Chromatic spectrum is broken

1999 ◽  
Vol 3 ◽  
pp. 86-89 ◽  
Author(s):  
Tao Jiang ◽  
Dhruv Mubayi ◽  
Zsolt Tuza ◽  
Vitaly Voloshin ◽  
Douglas West
Keyword(s):  
Chromatic Spectrum

scholarly journals Colouring Planar Mixed Hypergraphs

10.37236/1538 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
André Kündgen ◽  
Eric Mendelsohn ◽  
Vitaly Voloshin
Keyword(s):  
Planar Graph ◽  
Sharp Estimate ◽  
Chromatic Polynomial ◽  
Vertex Set ◽  
Chromatic Spectrum ◽  
Special Case ◽  
Mixed Hypergraph ◽  
Families Of Subsets

A mixed hypergraph is a triple ${\cal H} = (V,{\cal C}, {\cal D})\;$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, the ${\cal C}$-edges and ${\cal D}$-edges, respectively. A $k$-colouring of ${\cal H}$ is a mapping $c: V\rightarrow [k]$ such that each ${\cal C}$-edge has at least two vertices with a ${\cal C}$ommon colour and each ${\cal D}$-edge has at least two vertices of ${\cal D}$ifferent colours. ${\cal H}$ is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when ${\cal C}=\emptyset$ and all the ${\cal D}$-edges have size 2, whereas in a bi-hypergraph ${\cal C} = {\cal D}$. We investigate the colouring properties of planar mixed hypergraphs. Specifically, we show that maximal planar bi-hypergraphs are 2-colourable, find formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual, prove that their chromatic spectrum is gap-free and provide a sharp estimate on the maximum number of colours in a colouring.

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