Let δ(H) be the minimum degree of the graph H. We prove that a graph H of order n with δ(H) ⩾ (2n−1)/3 contains any graph G of order at most n and maximum degree Δ(G) ⩽ 2 as a subgraph, and this bound is best possible. Furthermore, this result settles the case Δ(G) = 2 of the well‐known conjecture of Bollobás, Catlin and Eldridge on packing two graphs with given maximum degree.
As a consequence of an early result of Pach we show that every maximal triangle-free
graph is either homomorphic with a member of a specific infinite sequence of graphs or
contains the Petersen graph minus one vertex as a subgraph. From this result and further
structural observations we derive that, if a (not necessarily maximal) triangle-free graph
of order n has minimum degree δ[ges ]n/3,
then the graph is either homomorphic with a
member of the indicated family or contains the Petersen graph with one edge contracted.
As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every
triangle-free graph with δ>n/3 is either homomorphic with
C5 or contains the Möbius
ladder. A major tool is the observation that every triangle-free graph with
δ[ges ]n/3 has a unique maximal triangle-free supergraph.
For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore's classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.
Graph Theory
International audience
In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.
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