For a graph G, if P (G) = { H 1 , H 2 , … , H t } is a set of pairwise disjoint subsets of V(G) such that V (G) = i = 1 t V (H i) and each subgraph induced by H i is k-connected, 1 ≤ i ≤ t , then P (G) is ca
Simultaneous representationscapture this in a very natural way. Givensimultaneous graphswhere each pair of graphsshare some common subgraph, asimultaneous representationasks for a fixed representation of each vertex that gives a valid representation of each. This notion is closely related topartial represe...
If G=(V,E) is a graph and X⊆V is a subset of vertices, we denote by G∖X the graph obtained from G by deleting all vertices in X. Equivalently, G∖X is the subgraph of G induced by V∖X, denoted by G[V∖X]. If X consists of a single vertex, say X={x}, we...
For a subgraph H of an edge-colored graph Gc, we use c(H) to denote the set of colors appearing on the edges of H. For two disjoint vertex subsets A and B of Gc, we use cG(A,B) to denote the set of colors of the edges between A and B in Gc, and use cG(v,B) instead...
We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a ...
For p ∈ N, a coloring A of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graphs...
Let Gi be the subgraph of G consisting of the first i ears of ED, that is, Gi = P1 ∪···∪ Pi. For each ear Pi, let si and ti be the endpoints of Pi. First, we present a claim concerning the lengths of Pi and some other paths in Gi. Claim 1. Let Pi be an ear from...
Builder is not allowed to draw an edge that would create a subgraph with edge density greater than d. Builder’s aim is to force Painter to create a monochromatic copy of H. The above Builder–Painter game has a different focus from the k-colouring game in the present paper. One ...
For a graph G and U⊆V(G), we denote by G[U] the subgraph of G induced by U, and G∖U=defG[V(G)∖U]. For a singleton {x} and a set Y, Y+x=defY∪{x} and Y−x=defY∖{x}. We use [n]=[1..n] to denote the set of positive integers less Bubble partitions...
It is demonstrated that any directed graph without loops and multiple edges can be embedded as an induced subgraph into the graph of essential dependence of some proper family of functions. Moreover, such embedding is reasonably economical, and the functions of the resulting proper family inherit ...