Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

An approach to edge bicoloring problems for biregular bipartite graphs is applied to graphs arising from cyclic groups as well as from odd graphs. In the case of the Petersen graph, this incidentally allows solutions to the Great Circle Challenge puzzle. On the other hand, for odd n = 2k+1 > 1, a one-to-one correspondence from the family of n-cycles of Kn onto the family Ok of n-cycles of the o...

We will characterize all graphs that have the property that the graph and its complement are minimal even pair free. This characterization allows a new formulation of the Strong Perfect Graph Conjecture. The reader is assumed to be familiar with perfect graphs (see for example [2]). A hole is a cycle of length at least five. An odd hole is a hole that has an odd number of vertices. An (odd) ant...

Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209–218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h...

Let G be a graph and f : V (G) → {1, 2, 3, . . . , p+ q} be an injection. For each edge e = uv and an integer m ≥ 2, the induced Smarandachely edge m-labeling f∗ S is defined by f ∗ S(e) = ⌈ f(u) + f(v) m ⌉ . Then f is called a Smarandachely super m-mean labeling if f(V (G))∪ {f∗(e) : e ∈ E(G)} = {1, 2, 3, . . . , p+ q}. Particularly, in the case of m = 2, we know that f ∗(e) =    f(u)+f(v) ...

The chordality of an undirected graph G, which is not acyclic, is defined as the length of the longest induced cycle in it. The chordality of an acyclic graph is defined to be 0. We use Cl (l ≥ 3) to denote a cycle of length l. An induced cycle is called a hole. A hole is an odd hole if its length is odd and is an even hole otherwise. Odd-chordality of a graph is the length of the longest odd h...

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f( ), then G is (2 + )-colorable. Note that the function f( ) is independent of the graph G and → 0 if and only if f( )→∞. A key lemma...

A signed graph is a pair (G,Σ) where G is a graph and Σ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of Σ. A blocking pair of (G,Σ) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts i...