‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

Given an undirected graph G = (V; E) and a partition fS; Tg of V , an S{T connector is a set of edges F E such that every component of the subgraph (V; F) intersects both S and T. If either S or T is a singleton, then an S{T connector is a spanning tree of G. On the other hand, if G is bipartite with colour classes S and T, then an S{T connector is an edge cover of G (a set of edges covering al...

In this paper some new results on the acyclic-edge coloring , f -edge coloring, g-edge cover coloring, (g, f )-coloring and equitable edge-coloring of graphs are introduced. In particular, some new results related to the above colorings obtained by us are given. Some new problems and conjectures are presented.

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequen...

The edge dominating set (EDS) and edge cover (EC) problems are classical graph covering problems in which one seeks a minimum cost collection of edges which covers the edges or vertices, respectively, of a graph. We consider the generalized partial cover version of these problems, in which failing to cover an edge, in the EDS case, or vertex, in the EC case, induces a penalty. The objective is ...

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of anti-ferromagnetic Ising model on line graphs is real rooted and roots edge cover have absolute value at most $4$. more generally show a $k$-uniform hypergraph $2^k$, discuss applications this domination polynomials graphs. moreover how our results relate efficient algorithms for approximately computin...

k≤m Ek and m is minimum. Two related and reasonably well-studied problems are the Minimum Biclique Cover (MBC), where Hk are required to be bicliques (i.e. complete bipartite subgraphs) [4,3,2,1] and the Minimum Cut Cover (MCC), where Hk are cutsets, namely not required to be connected. Both problems are NP-hard. To the best of our knowledge, whether the MBGC is NP-hard or not is currently unkn...

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best previously known hitting-set for this case had size (nw) where n is the number of variables and w is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We ...

The Shortest Cycle Cover Conjecture asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.4m. We show that every bridgeless graph with minimum degree three that contains m edges has a cycle cover comprised of three cycles of total length at most 44m/27 ≈ 1.6296m; this extends a bound of Fan [J. Graph Theory 18 (1994), 131–141] fo...