The clique partitioning problem (CPP) can be formulated as follows. Given is a complete graph G = (V; E), with edge weights w ij 2 R for all fi; jg 2 E. A subset A E is called a clique partition if there is a partition of V into non-empty, disjoint sets V 1 S k p=1 ffi; jgji; j 2 V p g. The weight of such a clique partition A is deened as P fi;jg2A w ij. The problem is now to nd a clique partit...

In this paper, we study structural properties of Toeplitz graphs. We characterize Kq-free graphs for an integer q≥3 and give equivalent conditions a graph Gn〈t1,t2,…,tk〉 with t1<⋯<tk n≥tk−1+tk being chordal Gn〈t1,t2〉 perfect. Then compute the edge clique cover number vertex graph. Finally, degree sequence (d1,d2,…,dn) n vertices show that is regular if only it circulant

Given a complete graph Kn = (V, E), with edge weight ce on each edge, we consider the problem of partitioning the vertices of graph Kn into subcliques that each have at least S vertices, so as to minimize the total weight of the edges that have both endpoints in the same subclique. It is an extension of the classic Clique Partition Problem that can be well solved using triangle inequalities, bu...

We study a signed variant of edge covers of graphs. Let b be a positive integer, and let G be a graph with minimum degree at least b. A signed b-edge cover of G is a function f : E(G) → {−1, 1} satisfying e∈EG(v) f(e) ≥ b for every v ∈ V (G). The minimum of the values of e∈E(G) f(e), taken over all signed b-edge covers f of G, is called the signed b-edge cover number and is denoted by ρb(G). Fo...

This paper examines a parameterized problem that we refer to as n− k Graph Coloring, i.e., the problem of determining whether a graph G with n vertices can be colored using n−k colors. As the main result of this paper, we show that there exists a O(kn+k+2) = O(n) algorithm for n− k Graph Coloring for each fixed k. The core technique behind this new parameterized algorithm is kernalization via m...

A graphs G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and is clique reducible if it is not clique irreducible. A graph G is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G and clique vertex reducible if it is not clique vertex irreducible. The clique vertex irred...