Fuzzy chromatic numberPerfect fuzzy graphPerfect fuzzy graph theoremPerfect graphs have a significant role in graph theory in terms of structure as well as application. This article embarked a comprehensive stud
Brooks' theorem: For any connected undirected graph G with maximum degree Δ, the chromatic number of G is at most Δ unless G is a complete graph or an odd cycle, in which case the chromatic number is Δ + 1.对于任何具有最大度数Δ的连通无向图G,除非G是一个完全图或一个奇圈,否则G的...
Theorem 2.3 If G is a graph with packing chromatic number j, and e is an edge of G, then j∕2+1≤χρ(Se(G))≤j+1. Moreover, for any k≥2 there exists a graph G with an edge e such that k=χρ(G)=χρ(Se(G))−1. Proof Given a packing coloring c of G, a packi...
Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.doi:10.1007/s00373-020-02197-2Gee-Choon LauHo-Kuen NgWai-Chee Shiu...
51. The edge chromatic number of join graph with fan and complete balanced bipartite graph was obtained. 52. The expressions of the chromatic function and the chromatic weight polynomial of a P-graph about a permutation group H are established, de Bruijn s Theorem follows from th. 53. One ...
Theorem 7.2 For any natural number n, the algebra E{n2+,31} is qualitatively repre- sentable on the graph K2n. Proof Immediate by Lemma 7.1. 8 Algebras E{n1+,31} These algebras are better known as Lyndon algebras. Lyndon [22] showed that for n ≥ 4, strong representations of the ...
This condition is then applied to the study of the circular chromatic number of My-cielski’s graphs,especially,the circular chromatic number of the iterated Mycielskian of complete graphs. For a graph G with vertex set V(G)=V and edge set E(G)=E,the Myciel-skian M(G)of G is ...
Theorem 1.2(a) is quite surprising, since already for planar graphs G, the exact distance graphs G[♮p] can be very dense. To see this, for i≥2 let Li be obtained from the complete graph K4 by subdividing each edge i−1 times (i.e. by replacing each edge by a path of lengt...
We now show that for the claimed values of ℓ the above bound for the expected number of colorings of the random graph Gn,p with ℓ colors tends to zero, i.e. χ(Gn,p)>ℓ holds with high probability. This will complete the proof of the theorem. Define the function αp(n)=...
COMPLETE graphsWe prove that for every path H $H$, and every integer d $d$, there is a polynomial f $f$ such that every graph G $G$ with chromatic number greater than f(t) $f(t)$ either contains H $H$ as an induced subgraph, or contains as a subgraph the complete d $d$‐...