Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k, 0)-colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 k+2 is (k, 0)-colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)-colorable. On the othe...

Labeling Subgraph Embeddings and Cordiality of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Ascending subgraph decompositions of regular graphs

We prove that every regular graph with ( n+1 2 ) + t edges, 06 t ¡ n + 1, can be decomposed into n subgraphs G1; G2; : : : ; Gn such that |E(Gi)| = i and Gi 6Gi+1 for i = 1; 2; : : : ; n − 1 and |E(Gn)|= n+ t. c © 2002 Elsevier Science B.V. All rights reserved.

The Maximum Acyclic Subgraph Problem and Degree-3 Graphs

Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find “as much of the pattern as possible” inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the patte...