Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.

We define classes of graphs based on forbidding and enforcing boundary conditions. Forbidding conditions prevent a graph to have certain combinations of subgraphs and enforcing conditions impose certain subgraph structures.We say that a class of graphs is an fe-class if the class can be defined through forbidding and enforcing conditions (fe-system). We investigate properties of fe-systems and ...

For an integer l¿0, de ne SE(l) to be the family of graphs such that G ∈SE(l) if and only if for any edge subset X ⊆E(G) with |X |6l, G has a spanning eulerian subgraph H with X ⊆E(H). The graphs in SE(0) are known as supereulerian graphs. Let f(l) be the minimum value of k such that every k-edge-connected graph is in SE(l). Jaeger and Catlin independently proved f(0) = 4. We shall determine f(...

Let G(V,E) be a graph of order n and size m. A graceful labeling of G is an injection f : V (G) → {0, 1, 2, ...,m} such that, when each edge uv is assigned the label f(uv) = |f(u)− f(v)|, the resultant edge labels are distinct. We focus on general results in graceful labeling, and provide an affirmative answer to the following open problem: Can every connected graph be embedded as an induced su...

The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...

We show that every 6-edge connected graph admits a circulation whose range lies in the interval 1; 3). The circular ow number c (G) of a ((nite) graph G is deened by c (G) = inffr 2 R: some orientation ~ G admits a circulation f : E(~ G) ! 1; r ? 1]g: This parameter is a reenement of the well studied ow number (G) := d c (G)e, which was introduced by Tutte as a dual to the chromatic number. Sin...

Algebraic techniques are used to find several new combinatorial interpretations for valuations of the Martin polynomial, M(G; s), for unoriented graphs. The Martin polynomial of a graph, introduced by Martin in his 1977 thesis, encodes information about the families of closed paths in Eulerian graphs. The new results here are found by showing that the Martin polynomial is a translation of a uni...

vVe prove that if G is a 2-edge-connected graph of order n 2: 14 and max{d{u),d(v)} > n3!) for each pair of nonadjacent vertices u~ v of G. then G contains a spanning Eulerian subgraph and hence the line graph of G is Hamiltonian.

Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.