A buttoning of a tree that has vertices v1, v2, . . . , vn is a closed walk that starts at v1 and travels along the shortest path in the tree to v2, and then along the shortest path to v3, and so forth, finishing with the shortest path from vn to v1. Inspired by a problem about buttoning a shirt inefficiently, we determine the maximum length of buttonings of trees

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted iₖ(G) and βₖ(G). We give some relations between βₖ(G) and ${\beta }_j\left(G\right)$ and between iₖ(G) and $i_j\left(G\right)$ for j ≠ k. We study two families of extremal graphs for the inequality i₂(G) ≤ i(G) + β(G). Finally we give an upper bound on i₂(G) and a lower bound when G is a cactus.

Given a graph $G$, let $f\left(G\right)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f\left(G\right)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. In this paper we prove that $f(m,{\theta }_{k,s})\ge \frac{1}{2}m+\Omega \left(m^{(2k+1)/(2k+2)}\right)$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K_k^{\text{'}}$ and $K_{t,s}^{\text{'}}$ for the subdivisions of $K_k$ and $K_{t,s}$. We show that $f(m,K_k^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5k-8)/(6k-10)}\right)$ and $f(m,K_{t,s}^{\text{'}})\ge \frac{1}{2}m+\Omega \left(m^{(5t-1)/(6t-2)}\right)$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs....

For a graph G = (V,E) and a vertex v ∈ V , let T(v) be a local trace at v, i.e. T(v) is an Eulerian subgraph of G such that every walk W(v), with start vertex v can be extended to an Eulerian tour in T(v). We prove that every maximum edge-disjoint cycle packing Z* of G induces a maximum trace T(v) at v for every v ∈ V . Moreover, if G is Eulerian then sufficient conditions are given that guarantee that the sets of cycles inducing maximum local traces of G also induce a maximum cycle packing of G....

An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) ${\chi }_r^{\text{'}}\left(G\right)$ is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 ${\chi }_r^{\text{'}}\left(G\right)\le 3$ for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i ${\chi }_1^{\text{'}}\left(G\right)=i$ , i...

A property of graphs is any class of graphs closed under isomorphism. Let ₁,₂,...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable if the vertex set V(G) can be partitioned into n sets, V₁,V₂,..., Vₙ, such that for each i = 1,2,...,n, the graph $G\left[V_i\right]{\in }_i$. We write ₁∘₂∘...∘ₙ for the property of all graphs which have a (₁,₂,...,ₙ)-partition. An additive induced-hereditary property is called reducible if there exist additive induced-hereditary properties ₁ and ₂ such that = ₁∘₂. Otherwise...

This paper investigates on those smallest regular graphs with given girths and having small crossing numbers.