A proper edge-k-coloring of a graph G is an assignment of k colors 1, 2, · · · , k to the edges of G such that no two adjacent edges receive the same color. A neighbor sum distinguishing edge-k-coloring of G is a proper edge-k-coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident with u is different from the sum of colors taken on the edges incident with...

We prove that the incidence chromatic number of every 3-degenerated graph G is at most ∆(G)+ 4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G) < 3 is at most ∆(G)+3. We show that when ∆(G) ≥ 5, this bound may be decreased to ∆(G) + 2. Moreover, we show that for every graph G with mad(G) < 22/9 (resp. with mad(G) < 16/7 and ∆(G) ≥ 4), this bou...

Suppose that I wish to make a graph G triangle-free by removing a small number of edges. An obvious obstruction is the presence of a large set of edge-disjoint triangles, since I must remove one edge from each triangle. On the other hand, removing all the edges in a maximal set of edge-disjoint triangles clearly makes G triangle-free. Tuza’s Conjecture states that the worstcase number of edges ...

Finding the largest independent set in a graph is a notoriously difficult NP -complete combinatorial optimization problem. Unlike other NP-complete problems, it does not admit a constant factor approximation algorithm for general graphs. Furthermore, even for graphs with largest degree 3, no polynomial time approximation algorithm exists with a 1.0071-factor approximation guarantee. We consider...

A proper vertex colouring of a graph G is 2-frugal (resp. linear) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph G is 2-frugally (resp. linearly) L-colourable if for a given list assignment L : V (G) → 2, there exists a 2-frugal (resp. linear) colouring c of G such that c(v) ∈ L(v) for all v ∈ V (G). If G is 2-frugally...

An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with maximum average degree less than 10 3 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least five has an oriented chromatic number at most 16, that improves the previous known bound of 19 due to ...

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree ∆ are list (∆ + 1)-...

The oriented chromatic number o(H) of an oriented graph H is de ned as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′. The oriented chromatic number o(G) of an undirected graph G is then de ned as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) de ned as the maximum average degree of the subgra...

in a graph g = (v,e), a subset s  v is said to be an open packing set if no two vertices of s have a common neighbour in g. the maximum cardinality of an open packing set is called the open packing number and is denoted by rho^o. this paper further studies on this parameter by obtaining some new bounds.

We show that Maximum Independent Set on connected graphs of average degree at most three can be solved in O(1.08537) time and linear space. This improves previous results on graphs of maximum degree three, as our connectivity requirement only functions to ensure that each connected component has average degree at most three. We link this result to exact algorithms of Maximum Independent Set on ...