A hypergraph
is said to be
-partite
-uniform if its vertex set
can be partitioned into non-empty sets
so that every edge in the edge set
, consists of precisely one vertex from each set
,
. It is ...denoted as
or
if
for
. In this paper we define
-partite self-complementary and almost self-complementary
-uniform hypergraph. We prove that, there exists an
-partite self-complementary
-uniform hypergraph
where
for
if and only if at least one of
is even. And we prove that, there exists an
-pasc
where
for
if and only if
are odd. Further, we analyze the cycle structure of complementing permutations of
-partite self-complementary
-uniform hypergraphs and
-partite almost self-complementary
-uniform hypergraphs.
In this paper, we study colorings of k-partite sparse digraphs. The chromatic number of a graph G is the smallest integer k such that the vertices of G can be colored with k colors with the property ...that each color class is an independent set. The dichromatic number of a digraph D is the minimum k such that the vertices of D can be colored with k colors with each color class inducing an acyclic subdigraph. This coloring invariant shares many similarities with the graph chromatic number and can be thought of as its analogous digraph generalization.
Our main result in this short note shows that there exist sparse k-partite digraphs which have dichromatic number k. This, in particular, not only implies that there exist graphs with equal chromatic and dichromatic number, but that they can be taken to be somewhat sparse.
In this paper we extend the notion of oriented regular representations in the context of m-partite oriented digraphs with an integer m≥2. A group G is said to admit an m-partite oriented semiregular ...representation (m-POSR for short) if there exists an m-partite oriented digraph such that its automorphism group is isomorphic to G and acts semiregularly on vertices with every part of the digraph as an orbit. In this paper, we proved that G admits an m-POSR of valency two except for several cases when G is a finite group generated by at most two elements and m≥2 is an integer.
In this paper we address the consensus problem in the context of networked agents whose communication graph splits into clusters: interactions between agents in the same cluster are cooperative, ...while interactions between agents belonging to different clusters are antagonistic. This problem set-up arises in the context of social networks and opinion dynamics, where reaching a consensus means that the opinions of the agents in the same cluster converge to the same decision, that is typically different for the different clusters. Under the assumption that agents belonging to the same cluster have the same amount of trust (/distrust) to be distributed among their cooperators (/adversaries), we propose a modified version of DeGroot’s law. By simply constraining how much agents in each group should be conservative about their own opinions, it is possible to achieve a nontrivial solution by means of a distributed algorithm. The result is then particularized to unweighted complete communication graphs, and subsequently extended to a class of nonlinear multi-agent systems.
We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil–Rödl Theorem, the Ramsey property of partial ...orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered S-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the structural Ramsey theorem. Both of these results are natural, and easy to state, yet their proofs involve most of the theory developed here.
We also characterise Ramsey lifts of classes of structures defined by finitely many forbidden homomorphisms and extend this to special cases of classes with closures. This has numerous applications. For example, we find Ramsey lifts of many Cherlin–Shelah–Shi classes.
We may view any graph as a network of resistors each having a resistance of 1 Ω. The resistance distance between a pair of vertices in a graph is defined as the effective resistance between the two ...vertices. This function is known to be a metric on the vertex-set of any graph. The main result of this paper is an explicit expression for the resistance distance between any pair of vertices in the complete n-partite graph Km1,m2,…,mn.
An equitable total coloring is the assignment of colors to the edges and vertices of a graph G so that incident and adjacent elements receive different colors, and the difference between the ...cardinalities of any two color classes is either 0 or 1. The least positive integer for which a graph G admits an equitable total coloring is called the graph’s equitable total chromatic number (denoted by χe′′). The Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002 states that Δ+1≤χe′′(G)≤Δ+2. Da Silva et al. (2018) verified Wang’s conjecture when G is a complete r-partite p-balanced graph, showing that χe′′=Δ+1 if G has odd order, χe′′=Δ+2 for r≥4 even and p odd and χe′′≤Δ+2 if G has even order for the other cases. In this work we improve this bound showing that χe′′=Δ+1 for all complete r-partite p-balanced graphs with p even. Moreover, since it is known that bipartite non balanced graphs have χe′′=Δ+1, we extend this study to infinite classes of tripartite complete non balanced graphs and show that all these graphs have equitable total chromatic number Δ+1.
Recent methods for object co-segmentation focus on discovering single co-occurring relation of candidate regions representing the foreground of multiple images. However, region extraction based only ...on low and middle level information often occupies a large area of background without the help of semantic context. In addition, seeking single matching solution very likely leads to discover local parts of common objects. To cope with these deficiencies, we present a new object co-segmentation framework, which takes advantages of semantic information and globally explores multiple co-occurring matching cliques based on an N-partite graph structure. To this end, we first propose to incorporate candidate generation with semantic context. Based on the regions extracted from semantic segmentation of each image, we design a merging mechanism to hierarchically generate candidates with high semantic responses. Second, all candidates are taken into consideration to globally formulate multiple maximum weighted matching cliques, which complement the discovery of part of the common objects induced by a single clique. To facilitate the discovery of multiple matching cliques, an N-partite graph, which inherently excludes intralinks between candidates from the same image, is constructed to separate multiple cliques without additional constraints. Further, we augment the graph with an additional virtual node in each part to handle irrelevant matches when the similarity between the two candidates is too small. Finally, with the explored multiple cliques, we statistically compute pixel-wise co-occurrence map for each image. Experimental results on two benchmark data sets, i.e., iCoseg and MSRC data sets achieve desirable performance and demonstrate the effectiveness of our proposed framework.
•The supercritical CO2 cooled in brazed plate heat exchangers is studied.•The brazed plate heat exchangers have diverse functions in tri-partite gas cooler.•The effects of different parameters on ...heat transfer and pressure drop are shown.•The well-known available heat transfer correlations for supercritical CO2 are evaluated.•The new heat transfer correlations for supercritical CO2 are developed.
The heat transfer characteristic of supercritical CO2 is an essential research topic due to its significant influence on the performance of heat exchangers and systems. In this paper, the heat transfer and pressure drop of supercritical CO2 in the brazed plate heat exchangers are experimentally researched. The heat exchangers belong to a tri-partite gas cooler which can simultaneously fulfill the demands of domestic hot water and space heating. The results demonstrates that the thermal resistance in the CO2 side is the main factor that influences the total heat transfer. The increase of CO2 inlet pressure can reduce the heat transfer coefficients except at the high temperature region. The improvement of heat transfer coefficient by increasing the CO2 mass flow rate is more significant in the space heating (SH) and domestic hot water (DHW) preheating gas coolers, and is lowest in the DHW reheating gas cooler. The influence of DHW inlet temperature is more obvious in the DHW preheating gas cooler that connected to the water inlet. The influence of water mass flow rate is different in the DHW and SH operation modes. Moreover, the effects of CO2 pressure and mass flow rate on the buoyancy force are discussed and the influence of buoyancy force on heat transfer is verified. The inaccuracy of the correlations from the literature is proved and then new correlations are established. The mean absolute relative errors of the new correlations are 11.61% and 12.82% for the one-pass and two-pass configurations, respectively. Furthermore, the frictional pressure drop in the heat exchangers is low (up to 36.51 kPa) and basically increases as the Reynolds number increases.
•Explore k-partite entanglement and k-nonseparability of general N-partite quantum states.•Present some simple and powerful k-partite entanglement and k-nonseparability criteria.•Our criteria perform ...better than other known detection criteria.•Detailed examples are provided.
Identifying the k-partite entanglement and k-nonseparability of general N-partite quantum states is a fundamental issue in quantum information theory. By use of computable inequalities of nonlinear operators, we present some simple and powerful k-partite entanglement and k-nonseparability criteria that work very well and allow for a simple and inexpensive test for the whole hierarchy of k-partite entanglement and k-nonseparability of N-partite systems with k running from N down to 2. We illustrate their strengths by considering several examples in which our criteria perform better than other known detection criteria. We are able to detect k-partite entanglement and k-nonseparability of multipartite systems which have previously not been identified. In addition, our results can be implemented in today's experiments.
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