A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 73 admits a 2-distance (Δ + 1)-coloring. This result is tight. This improves previous known results of Dolama and Sopena.

This paper theoretically and empirically studies the degree and connectivity of the Internet’s scale-free topology at an autonomous system (AS) level. The basic features of scale-free networks influence the normalization constant of degree distribution p(k). It develops a new mathematic model for describing the power-law relationships of Internet topology. From this model we theoretically obtai...

Let a, b, d be non-negative integers. A graph G is (d, a, b)-colorable if its vertex set can be partitioned into a + b sets D1, . . . , Da, O1, . . . , Ob such that the graph G[Di] induced by Di has maximum degree at most d for 1 ≤ i ≤ a, while the graph G[Oj ] induced by Oj is an edgeless graph for 1 ≤ j ≤ b. In this paper, we give two real-valued functions f and g such that any graph with max...

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+o(1)) √ m where m is the maximum degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm3−β for some constant c ...

The square G of a graph G is the graph obtained from G by adding an edge between every pair of vertices having a common neighbor. A proper coloring of G is also called a 2-distance coloring of G. The maximum average degree Mad(G) of a graph G is the maximum among the average degrees of the subgraphs of G, i.e. Mad(G) = max { 2|E(H)| V (H) |H ⊆ G } . Graphs with bounded maximum average degree ar...