Let G be a directed graph embedded in a surface. A map φ : E(G) → R is a tension if for every circuit C ⊆ G, the sum of φ on the forward edges of C is equal to the sum of φ on the backward edges of C. If this condition is satisfied for every circuit of G which is a contractible curve in the surface, then φ is a local tension. If 1 ≤ |φ(e)| ≤ α − 1 holds for every e ∈ E(G), we say that φ is a (l...

we give a new recursive method to compute the number of cliques and cycles of a graph. this method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. in particular, let $g$ be a graph and let $overline {g}$ be its complement, then given the chromatic polynomial of...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

A k-edge colouring (not necessarily proper) of a graph with colours in {1,2,…,k} is neighbour sum distinguishing if, for any two adjacent vertices, the sums edges incident each them are distinct. The smallest value k such that G exists denoted by χ∑e(G). When we add additional restriction must be proper, then χ∑′(G). Such colourings studied on connected at least 3 vertices. There famous conject...

We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with ∆ ≤ 5, but polynomial on bipartite graphs with ∆ ≤ 3, for which we construct an O(n)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bip...

Following [1], we investigate the problem of covering a graph G with induced subgraphs G1, . . . , Gk of possibly smaller chromatic number, but such that for every vertex u of G, the sum of reciproquals of the chromatic numbers of the Gi’s containing u is at least 1. The existence of such “chromatic coverings” provides some bounds on the chromatic number of G.