The distance-3 graph S3 of the Biggs-Smith graph S is shown to be: (a) a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-configuration; (b) a union of 102 cuboctahedra, (copies of L(Q3)), with no 2 such cuboctahedra having a common chordless 4-cycle; (c) not a line graph. Moreover, S3 is shown to have a C-ultr...

In this paper, we study the crossing number of the complete bipartite graph K4,n in torus and obtain crT (K4,n) = ⌊ n 4 ⌋(2n− 4(1 + ⌊ n 4 ⌋)).

The WD40 repeat protein WDR5 specifically associates with the K4-methylated histone H3 in human cells. To investigate the structural basis for this specific recognition, we have determined the structure of WDR5 in complex with a dimethylated H3-K4 peptide at 1.9 A resolution. Unlike the chromodomain that recognizes the methylated H3-K4 through a hydrophobic cage, the specificity of WDR5 for met...

A connected edge-disjoint union Y of 102 tetrahedra (copies ofK4) is constructed which is Menger graph of a self-dual (1024)-configuration and K4-ultrahomogeneous. As Y is not a line graph, we ask whether there exists a non-line-graphical K4-ultrahomogeneous Menger graph of a self-dual (n4)-configuration which is connected edge-disjoint union of n copies of K4, for n / ∈ {42, 102}. Moreover, Y ...

The richest class of t-perfect graphs known so far consists of the graphs with no so-called odd-K4. Clearly, these graphs have the special property that they are hereditary t-perfect in the sense that every subgraph is also t-perfect, but they are not the only ones. In this paper we characterize hereditary t-perfect graphs by showing that any non–t-perfect graph contains a non–tperfect subdivis...

We study the mixed Ramsey number maxR(n,Km,Kr), defined as the maximum number of colours in an edge-colouring of the complete graph Kn, such that Kn has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,Km,K4) ≤ n3/2 √ 2m for all m ≥ 3. Further, we discuss a possible way to impr...

The Ramsey number R(Cn or Kn1 , Km) is the smallest integer p such that every graph G on p vertices contains either a cycle en with length n or a K n 1 , or an independent set of order m. In this paper we prove that R(Cn or K n b K 3 ) = 2(n 2) + 1 (n 2: 5), R(Cn or K n 1 , K4) = 3(n 2) + 1 (n 2: 7). In particular, we prove that R(C4 or K 3 , K 3 ) = 6, R(C4 or K 3 , K4) = 8, R(C5 or K4, K 4 ) ...