215 research outputs found
Chromaticity of a family of 5-partite graphs
AbstractLet P(G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted GâŒH, if P(G,λ)=P(H,λ). We write [G]={HâŁHâŒG}. If [G]={G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 5-partite graphs G with 5n vertices according to the number of 6-independent partitions of G. Using these results, we investigate the chromaticity of G with certain stars or matching deleted parts . As a by-product, two new families of chromatically unique complete 5-partite graphs G with certain stars or matching deleted parts are obtained
Diszkrét matematika = Discrete mathematics
A pĂĄlyĂĄzat rĂ©sztvevĆi igen aktĂvak voltak a 2006-2008 Ă©vekben. Nemcsak sok eredmĂ©nyt Ă©rtek el, miket több mint 150 cikkben publikĂĄltak, eredmĂ©nyesen nĂ©pszerƱsĂtettĂ©k azokat. Több mint 100 konferenciĂĄn vettek rĂ©szt Ă©s adtak elĆ, felerĂ©szben meghĂvott, vagy plenĂĄris elĆadĂłkĂ©nt. HagyomĂĄnyos grĂĄfelmĂ©let Több extremĂĄlis grĂĄfproblĂ©mĂĄt oldottunk meg. Ăj eredmĂ©nyeket kaptunk Ramsey szĂĄmokrĂłl, globĂĄlis Ă©s lokĂĄlis kromatikus szĂĄmokrĂłl, Hamiltonkörök lĂ©tezĂ©sĂ©sĂ©rĆl. a crossig numberrĆl, grĂĄf kapacitĂĄsokrĂłl Ă©s kizĂĄrt rĂ©szgrĂĄfokrĂłl. VĂ©letlen grĂĄfok, nagy grĂĄfok, regularitĂĄsi lemma Nagy grĂĄfok "hasonlĂłsĂĄgait" vizsgĂĄltuk. KĂŒlönfĂ©le metrikĂĄk ekvivalensek. Ć°j eredemĂ©nyeink: Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit. HipergrĂĄfok, egyĂ©b kombinatorika Ăj Sperner tipusĂș tĂ©telekte kaptunk, aszimptotikusan meghatĂĄrozva a halmazok max szĂĄmĂĄt bizonyos kizĂĄrt struktĆrĂĄk esetĂ©n. Több esetre megoldottuk a kizĂĄrt hipergrĂĄf problĂ©mĂĄt is. ElmĂ©leti szĂĄmĂtĂĄstudomĂĄny Ăj ujjlenyomat kĂłdokat Ă©s bioinformatikai eredmĂ©nyeket kaptunk. | The participants of the project were scientifically very active during the years 2006-2008. They did not only obtain many results, which are contained in their more than 150 papers appeared in strong journals, but effectively disseminated them in the scientific community. They participated and gave lectures in more than 100 conferences (with multiplicity), half of them were plenary or invited talks. Traditional graph theory Several extremal problems for graphs were solved. We obtained new results for certain Ramsey numbers, (local and global) chromatic numbers, existence of Hamiltonian cycles crossing numbers, graph capacities, and excluded subgraphs. Random graphs, large graphs, regularity lemma The "similarities" of large graphs were studied. We show that several different definitions of the metrics (and convergence) are equivalent. Several new results like the Hereditary Property Testing, Inverse Counting Lemma and the Uniqueness of Hypergraph Limit were proved Hypergraphs, other combinatorics New Sperner type theorems were obtained, asymptotically determining the maximum number of sets in a family of subsets with certain excluded configurations. Several cases of the excluded hypergraph problem were solved. Theoretical computer science New fingerprint codes and results in bioinformatics were found
Graphs determined by polynomial invariants
AbstractMany polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results
Chromatic Equivalence Classes and Chromatic Defining Numbers of Certain Graphs
There are two parts in this dissertation: the chromatic equivalence classes and
the chromatic defining numbers of graphs.
In the first part the chromaticity of the family of generalized polygon trees with
intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that
Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} â„ r+3. We consider
Cr( a, b; c, d) when min{ a, b, c, d} †r + 2. The necessary and sufficient conditions
for Cr(a, b; c, d) with min {a, b, c, d} †r + 2 to be a chromatic equivalence class
are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized.
In the second part the defining numbers of regular graphs are studied. Let
d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs
of order n and the chromatic number equals to k. It is proved that for a given
integer k and each r â„ 2(k - 1) and n â„ 2k, d(n, r, X = k) = k - 1. Next,
a new lower bound for the defining numbers of r-regular k-chromatic graphs
with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when
k < r < 2(k - 1) for certain values of n and r is determined
Chromatic equivalence class of the join of certain tripartite graphs
For a simple graph G, let P(G;λ) be the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent, denoted G ~ H if P(G;λ) = P(H;λ). A graph G is said to be chromatically unique, if H ~ G implies that H â
G. Chia [4] determined the chromatic equivalence class of the graph consisting of the join of p copies of the path each of length 3. In this paper, we determined the chromatic equivalence class of the graph consisting of the join of p copies of the complete tripartite graph K1,2,3. MSC: 05C15;05C6
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