Homomorphism complexes were introduced by Lovász to study topological obstructions to graph colorings. We show that folding in the second parameter of the homomorphism complex yields a homotopy equivalence. We study how the Mycielski construction changes the homotopy type of the homomorphism complex. We construct graphs showing that the topological bound obtained by odd cycles can be arbitraril...

We prove that the coindex of the box complex B(H) of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph H a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then coind(B(H))+2 ≤ 3; if the system has solutions, then χ(H) ≥ 4. We generalis...

Let H be a bipartite graph and let Gn be the Mycielski graph with χ(G) = n, n ≥ 4. Then the chromatic number of the strong product of Gn by H is at most 2n− 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H, any triangles of G trans...

In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in [20] that t Mycielskian of the Kneser graph KG(m,n) has the same circular chromatic number and chromatic number provided that m + t is an even integer. We prove that if m is large enough, then χ(M (KG(m,n))) = χc(M (KG(m,n))) where M t is t Mycielskian. Also, we consider the generalized ...

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G. We first define the operation μ(G) and the...

The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m,n, r, s) such that each m × n 0-1 matrix containing Z(m,n, r, s) ones has an r × s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m, n, r, s) such that each bipartite graph on m black vertices and n white vertices with Z(m,n, r, ...

We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that 1 and 0 occur equally often in it. Our contribution is twofold. First, we study weaker forms of the conjecture proved in the past and lay out detailed proof...

Recenzja edycji: Zygmunt Mycielski, znaki zapytania. Wstęp i komentarze Beata Bolesławska-Lewandowska, red. Marek Zagańczyk, Kraków 2022.