Maximum Degree Of Graph Research Articles

The maximum size of an edge 2-neighborhood in P5-free graphs

For a graph G=(V,E) and an edge uv∈E(G), the 2-neighborhood of uv is the set of all edges having at least one endvertex in N(u)∪N(v). A graph is called P5-free if it contains no induced subgraphs isomorphic to a path with 5 vertices. For P5-free graphs, we show that the maximum cardinality of an edge 2-neighborhood is at most 5Δ24, where Δ is the maximum degree of graphs. When Δ is even, this bound is tight and confirms the strong edge coloring conjecture of Erdős and Nešetřil for P5-free graphs.

Approximation algorithm for (connected) bounded-degree deletion problem on unit disk graphs

In this paper, we study the minimum (connected) k-bounded-degree node deletion problem (Min(C)kBDND). For a connected graph G, a constant k and a weight function w:V→R+, a vertex set C⊆V(G) is a kBDND-set if the maximum degree of graph G−C is at most k. If furthermore, the subgraph of G induced by C is connected, then C is a CkBDND-set. The goal of MinWkBDND (resp. MinWCkBDND) is to find a kBDND-set (resp. CkBDND-set) with the minimum weight. In this paper, we focus on their cardinality versions with w(v)≡1,v∈V, which are denoted as MinkBDND and MinCkBDND. This paper presents a (1+ε) and a 3.76-approximation algorithm for MinkBDND and MinCkBDND on unit disk graphs, respectively, where 0<ε<1 is an arbitrary constant.

Relation between the Hermitian energy of a mixed graph and the matching number of its underlying graph

ABSTRACT Given a graph G, the mixed graph is obtained from G by orienting some of its edges (G is also called the underlying graph of ). Let be the Hermitian energy of and let be the matching number of the underlying graph G. In this paper, we first establish the inequality and characterize all the mixed graphs which make hold. Furthermore, we obtain , where and are, respectively, the vertex cover number, the number of odd cycles and the maximum degree of graph G. The lower bound is attained if and only if is switching equivalent to its underlying graph G, where G is the disjoint union of some complete bipartite graphs each of which contains a perfect matching, together with some isolated vertices. The upper bound is best possible. By our results in this paper, some main results in Tian FL, Wong D. [Relation between the skew energy of an oriented graph and its matching number. Discrete Appl Math. 2017;222:179–184]; Wang L, Ma XB. [Bounds of graph energy in terms of vertex cover number. Linear Algebra Appl. 2017;517:207–216]; Wong D, Wang XL, Chu R. [Lower bounds of graph energy in terms of matching number. Linear Algebra Appl. 2018;549:276–286] can be deduced in a unified approach.

An approximation algorithm for maximum weight budgeted connected set cover

This paper studies approximation algorithm for the maximum weight budgeted connected set cover (MWBCSC) problem. Given an element set $$X$$X, a collection of sets $${\mathcal {S}}\subseteq 2^X$$S⊆2X, a weight function $$w$$w on $$X$$X, a cost function $$c$$c on $${\mathcal {S}}$$S, a connected graph $$G_{\mathcal {S}}$$GS (called communication graph) on vertex set $${\mathcal {S}}$$S, and a budget $$L$$L, the MWBCSC problem is to select a subcollection $${\mathcal {S'}}\subseteq {\mathcal {S}}$$S?⊆S such that the cost $$c({\mathcal {S'}})=\sum _{S\in {\mathcal {S'}}}c(S)\le L$$c(S?)=?S?S?c(S)≤L, the subgraph of $$G_{\mathcal {S}}$$GS induced by $${\mathcal {S'}}$$S? is connected, and the total weight of elements covered by $${\mathcal {S'}}$$S? (that is $$\sum _{x\in \bigcup _{S\in {\mathcal {S'}}}S}w(x)$$?x??S?S?Sw(x)) is maximized. We present a polynomial time algorithm for this problem with a natural communication graph that has performance ratio $$O((\delta +1)\log n)$$O((?+1)logn), where $$\delta $$? is the maximum degree of graph $$G_{\mathcal {S}}$$GS and $$n$$n is the number of sets in $${\mathcal {S}}$$S. In particular, if every set has cost at most $$L/2$$L/2, the performance ratio can be improved to $$O(\log n)$$O(logn).

Relationships between total domination, order, size, and maximum degree of graphs

AbstractA total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γt(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6, it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γt(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γt(G))/2 holds. In this article, we prove that $\leq (\Delta(G)+ 2\sqrt{\Delta}) (|V(G)| - \gamma_{t}(G))/2$ holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γt(G))/2. The last result also disproves a conjecture on domination numbers from 8. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007