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 discuss a...

We introduce the class of outerspatial 2-complexes as natural generalisation outerplanar graphs to three dimensions. Answering a question O-joung Kwon, we prove that locally 2-connected 2-complex is if and only it does not contain surface positive genus subcomplex have space minor generalised cone over $K_4$ or $K_{2,3}$.
 This applied nested plane embeddings graphs; is, constrained by con...

The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3-colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3-color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3,K3 + e,K4 − e,K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey ...

Rhombellanes are mathematical structures existing in various environments, in crystal or quasicrystal networks, or even in their homeomorphs, further possible becoming real molecules. Rhombellanes originate in the K2.3 complete bipartite graph, a tile found in the linear polymeric staffanes. In close analogy, a rod-like polymer derived from hexahydroxy-cyclohexane was imagined. Further, the ide...

A vertex of a graph is bisimplicial if the set its neighbors union two cliques; quasi-line every bisimplicial. recent result Chudnovsky and Seymour asserts that non-empty even-hole-free has vertex. Both Hadwiger's conjecture Erdős-Lovász Tihany have been shown to be true for graphs, but are open graphs. In this note, we prove $G$ with $\omega(G)<\chi(G)=s+t-1$ satisfies provided $ t\geq s &g...

We present a fully computer-assisted proof system for solving particular family of problems in Extremal Combinatorics. Existing techniques using Flag Algebras have proven powerful the past, but so far lacked computational counterpart to derive matching constructive bounds. demonstrate that common search heuristics are capable finding constructions beyond reach human intuition. Additionally, mos...

Turán's famous tetrahedron problem is to compute the Turán density of K 4 3 $K_4^3$ . This equivalent determining maximum ℓ 1 $\ell _1$ -norm codegree vector a -free n $n$ -vertex 3-uniform hypergraph. We introduce new way for measuring extremality hypergraphs and determine asymptotically extremal function in our notion. The squared sum, co 2 ( G ) $\mbox{co}_2(G)$ , hypergraph $G$ sum codegree...