The k-Coloring problem is to test whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. If a graph G does not contain a graph H as an induced subgraph, then G is called H-free. For any fixed graph H on at most 6 vertices, it is known that 3-Coloring is polynomial-time solvable on H-free graphs whenever H is a linear forest and NP-complet...

The bandwidth problem has a long history and a number of important applications. It is the problem of enumerating the vertices of a given graph G such that the maximum difference between the numbers of adjacent vertices is minimal. We will show for any constant k 2 IN that there is no polynomial time approximation algorithm with an approximation factor of k. Furthermore, we will show that this ...

In this paper, we study the Minimum Sum Coloring (MSC) problem on P4-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. Based in the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph, we show that...

We study the minimum number of weights assigned to the edges of a graph G with no component K2 so that any two adjacent vertices have distinct sets of weights on their incident edges. The best possible upper bound on this parameter is proved.

We show that the number of points with pairwise different sets of neighbors in a graph is O(2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound.

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs wh...

In 2012 Seager introduced a new variant of the Cops and Robbers game, in which a cop searches for a moving robber on a graph using distance probes. The robber on his turn can either remain still or move to an adjacent vertex, while the cop on her turn probes any vertex and learns the robber’s distance from the probed vertex. Carraher, Choi, Delcourt, Erickson and West later showed that for any ...

Residents of socioeconomically deprived areas perceive their neighbourhood as less conducive to healthy behaviours than residents of more affluent areas. Whether these unfavourable perceptions are based on objective neighbourhood features or other factors is poorly understood. We examined individual and contextual correlates of socioeconomic inequalities in neighbourhood perceptions across five...

the topological index of a graph g is a numeric quantity related to g which is invariant underautomorphisms of g. the vertex pi polynomial is defined as piv (g)  euv nu (e)  nv (e).then omega polynomial (g,x) for counting qoc strips in g is defined as (g,x) =cm(g,c)xc with m(g,c) being the number of strips of length c. in this paper, a new infiniteclass of fullerenes is constructed. the ...

In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The t-pebbling number πt(G) of a graph G is the smallest m such that for every initial distribution of m pebbles on V (G) and every target vertex x there exists a sequence of pebbling moves leading t...