Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.
Let G and H be simple graphs. The Ramsey number r(G, H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of K r contains a red subgraph G or a blue subgraph H. The size Ramsey numberr(G, H) for a pair of graphs G and H is the smallest numberr such that there exists a graph F with sizer satisfying the property that any red-blue coloring of the edges of F contains a red subgraph G or a blue subgraph H. Additionally, if the order of F in the size Ramsey number equals r(G, H), then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey numbers for pairs of small graphs with orders at most four. continued Harary and Miller's works and summarized the complete results on the (restricted) size Ramsey numbers for pairs of small graphs with orders at most four. In 1998, Lortz and Mengenser gave both the size Ramsey numbers and the restricted size Ramsey numbers for pairs of small forests with orders at most five. To continue their works, we investigate the restricted size Ramsey numbers for a path of order three versus any connected graph of order five.
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>
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