Let G = be a graph, with and . An injective mapping is called an even-odd harmonious labeling of the graph G, if induced edge such that (i) bijective (ii) The acquired from this graph. In paper, we discovered some interesting results like H-graph, comb bistar for labeling.

We present a proof, using spectral techniques, that there is no finite measurable coloring of the odd-distance graph.

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy ai,j = ai−1,j−1+ai−1,j for all 2 6 i < j 6 n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states...

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, an...

A weak odd dominated (WOD) set in a graph is a subset B of vertices such that ∃D ⊆ V \B, ∀v ∈ B, |N(v)∩D| = 1 mod 2. We point out the connections of weak odd domination with odd domination, (σ, ρ)-domination, and perfect codes. We introduce bounds on κ(G), the maximum size of WOD sets of a graph G, and on κ(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding dec...

We prove that a minimal imperfect graph containing a vertex which is not on any P 5 has no odd pair. A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by (H)) not exceeding the cardinality !(H) of a maximum clique of H. A graph is minimal imperfect if all its proper in...

In this work some new odd graceful graphs are investigated. We prove that the graph obtained by fusing all the n vertices of Cn of even order with the apex vertices of n copies of K1,m admits odd graceful labeling. In addition to this we show that the shadow graphs of path Pn and star K1,n are odd graceful graphs.

Sewell and Trotter proved that every connected α-critical graph that is not isomorphic to K1,K2 or an odd cycle contains a totally odd K4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter’s theorem.

in this paper we define the removable cycle that, if $im$ is a class of graphs, $gin im$, the cycle $c$ in $g$ is called removable if $g-e(c)in im$. the removable cycles in eulerian graphs have been studied. we characterize eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for eulerian graph to have removable cycles h...

All K4-free graphs with no odd hole and no odd antihole are three-colourable, but what about K4free graphs with no odd hole? They are not necessarily three-colourable, but we prove a conjecture of Ding that they are all four-colourable. This is a consequence of a decomposition theorem for such graphs; we prove that every such graph either has no odd antihole, or belongs to one of two explicitly...