Suppose F, G, and H be simple graphs. We let F → (G, H) denote red-blue coloring of edges of F containing either red G or blue H. The graph F is considered Ramsey (G, H)-minimal if F → (G, H) and F − e ↛ (G, H) for arbitrary edge e of E(F ). The set of (G, H)-minimal graphs is denoted by R(G, H). In this paper, we study an infinite family of graphs belongs to R(K
1,2, C
4).
<p>Suppose <em>G</em>(<em>V,E</em>) be a connected simple graph and suppose <em>u,v,x</em> be vertices of graph <em>G</em>. A bijection <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,|<em>V</em> (<em>G</em>)| + |<em>E</em>(<em>G</em>)|} is called super local edge antimagic total labeling if for any adjacent edges <em>uv</em> and <em>vx</em>, <em>w</em>(<em>uv</em>) 6= <em>w</em>(<em>vx</em>), which <em>w</em>(<em>uv</em>) = <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) for every vertex <em>u,v,x</em> in <em>G</em>, and <em>f</em>(<em>u</em>) < <em>f</em>(<em>e</em>) for every vertex <em>u</em> and edge <em>e</em> ∈ <em>E</em>(<em>G</em>). Let γ(<em>G</em>) is the chromatic number of edge coloring of a graph <em>G</em>. By giving <em>G</em> a labeling of <em>f</em>, we denotes the minimum weight of edges needed in <em>G</em> as γ<em>leat</em>(<em>G</em>). If every labels for vertices is smaller than its edges, then it is be considered γ<em>sleat</em>(<em>G</em>). In this study, we proved the γ sleat of paths and its derivation.</p>
Let G(V, E) be a simple graph and f be a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} where f (V ) = {1, 2, . . . , |V |}. For a vertex x ∈ V , define its weight w(x) as the sum of labels of all edges incident with x and the vertex label itself. Then f is called a super vertex local antimagic total (SLAT) labeling if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χ slat (G) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. We classify all trees T that have χ slat (T ) = 2, present a class of trees that have χ slat (T ) = 3, and show that for any positive integer n ≥ 2 there is a tree T with χ slat (T ) = n.
Let G = (V, E) be a simple undirected graph. A labeling f : V (G) → {1, . . . , k} is a local inclusive d-distance vertex irregular labeling of G if every adjacent vertices x, y ∈ V (G) have distinct weights, with the weight w(x), x ∈ V (G) is the sum of every labels of vertices whose distance from x is at most d. The local inclusive d-distance vertex irregularity strength of G, lidis(G), is the least number k for which there exists a local inclusive d-distance vertex irregular labeling of G. In this paper, we prove a conjecture on the local inclusive d-distance vertex irregularity strength for d = 1 for tree and we generalize the result for block graph using the clique number. Furthermore, we present several results for multipartite graphs and we also observe the relationship with chromatic number.
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