If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number
of vertices of $G$ and $Delta=Delta(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $Delta$-colorable if $G$ satisfies $(|G|+1)/3 leq Delta < |G|/2$ and none of its components is a $K_{Delta +1}$.
A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove that a conne
cted graph $G$ is near-factor-critical if and only if it has a perfect matching. We also characterize disconnected near-factor-critical graphs.
Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square colo
ring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing colori
ng of G is denoted by $chi_a(G)$. In this paper, we prove that $chi_a(G)$ <= 5($Delta+2$)/2 for any graph G having maximum degree $Delta$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $chi_a(G)$ <= 3$Delta$ for any graph G without isolated edges.
Let D be an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let m(G) denote the minimum number of dependent arcs over all acyclic orientations of G. For any k > 0, a generalized Mycielski graph M
_k(G) of G is defined. Note that M_1(G) is the usual Mycielskian of G. We generalize results concerning m(M_1(G)) in K. L. Collins, K. Tysdal, J. Graph Theory, 46 (2004), 285-296, to m(M_k(G)). The underlying graph of a Hasse diagram is called a cover graph. Let c(G) denote the the minimum number of edges to be deleted from a graph G to get a cover graph. Analogue results about c(G) are also obtained.
Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all
acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t
): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let m and M denote the minimum and the maximum of the number of dependent arcs over all acyclic orientations of G. We cal
l G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable.
