A graph G contains a subdivision of H if G contains a subgraph which is isomorphic to a graph that can be obtained from H by subdividing some edges. A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. Althou...

Haj\'os conjectured that every graph containing no subdivision of the complete $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, maximum chromatic number such graphs $\Omega(s^2/\log s)$. We prove $O(s)$ colors are enough for a weakening this only requires monochromatic component to have bounded size (so-called clustered coloring). Our approach leads more res...

We show that for every ε > 0 there exists an r0 = r0(ε) such that for all integers r ≥ r0 every graph of average degree at least r + ε and girth at least 1000 contains a subdivision of Kr+2. Combined with a result of Mader this implies that for every ε > 0 there exists an f(ε) such that for all r ≥ 2 every graph of average degree at least r + ε and girth at least f(ε) contains a subdivision of ...

We prove that the γ-vector of the barycentric subdivision of a simplicial sphere is the f -vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.

A well known theorem of Kuratowski states that a graph is planar iff it contains no subdivision of K5 or K3,3. Seymour conjectured in 1977 that every 5-connected nonplanar graph contains a subdivision of K5. In this paper, we prove several results about independent paths (no vertex of a path is internal to another), which are then used to prove Seymour’s conjecture for two classes of graphs. Th...

We prove that every internally 4-connected non-planar bipartite graph has an odd K3,3 subdivision; that is, a subgraph obtained from K3,3 by replacing its edges by internally disjoint odd paths with the same ends. The proof gives rise to a polynomial-time algorithm to find such a subdivision. (A bipartite graph G is internally 4-connected if it is 3-connected, has at least five vertices, and th...

Shary considers the method “to turn out better than the traditional techniques from [11, 6, 8] in either the computational efficacy and the quality of the results it produces”. Although Shary presents it as an isolated phenomenon, it turns out that the graph subdivision method is intimately intertwined with a variety of methods in optimization. It is the purpose of this paper to present a unifi...

A graph G is said to have property Pm if it contains no subdivision of Km+1 and no subdivision of Kdm=2e+1;bm=2c+1. Chartrand et al. (J. Combin Theory 10 (1971) 12–41) (see also Problem 6.3 in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995) conjectured that the set of vertices (respectively, edges) of any graph with property Pm can be partitioned into m−n+1 subsets such that ea...

A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

Midpoint subdivision generalizes the Lane-Riesenfeld algorithm for uniform tensor product splines and can also be applied to non regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo-Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull-Clark algorithm. In 2001, Zorin and Schröder were able to prove C1-continuity for midpoint subdivision surfaces a...