A Collection of Minimally Path Square-Saturated Graphs

Partitions of graphs and multigraphs under degree constraints

In this paper, we show the following: Let G be a connected claw-free graph such that G has a connected induced subgraph H that has a pair of vertices $$\{v_{1}, v_{2}\}$$ of degree one in H whose distance is $$d + 2$$ in H. Then H has an induced subgraph F, which is isomorphic to $$B_{i,j}$$ , with $$\{v_{1}, v_{2}\} \subseteq V(F)$$ and $$i+j=d+1$$ , with a well-defined exception. Here $$B_{i, j}$$ denotes the graph obtained by attaching two vertex-disjoint paths of lengths $$i, j \ge 1$$ to a triangle. We also use the result above to strengthen the results in Xiong et al. (Discrete Math 313:784–795, 2013) in two cases, when $$i + j \le 9$$ , and when the graph is $$\Gamma _{0}$$ -free. Here $$\Gamma _{0}$$ is the simple graph with degree sequence 4, 2, 2, 2, 2. Let $$i, j > 0$$ be integers such that $$i + j \le 9$$ . Then The two results above are all sharp in the sense that the condition “ $$i+j\le 9$$ ” couldn’t be replaced by $$``i+j\le 10$$ ”.

Conditions for [formula omitted]-graphic sequences to be potentially [formula omitted]-graphic

Spectra of quasi-strongly regular graphs

Degree-based graph construction is a fundamental problem in network science. A graph is simple if there are no self-loops and no multiple links between any pair of nodes in the graph. A degree sequence d = ( d 1 , d 2 , ⋯ , d N ) is graphical if d can be represented as the degree sequence of at least one simple graph, where the graph is called a realization of the sequence d. In this work, we introduce a novel method (LSFGR) for generating simple graphs from graphical degree sequences, focusing additionally on connectedness and on assortativity. LSFGR guarantees connected graphs for all potentially connected degree sequences. In the case where a degree sequence has no simple realization, LSFGR produces graphs with at most one node with self-loops. In addition, the graphs generated from LSFGR characterize real-world networks with medium assortativity.

Rainbow colouring of split graphs

In this article, we obtain some Ore‐type sufficient conditions for a graph to have a connected factor with degree restrictions. Let α and k be positive integers with $\alpha \ge {{k + 1} \over{k - 1}}$ if ${{k}} \ge 2$ and $\alpha \ge 4$ if ${{k}}=1$. Let G be a connected graph with a spanning subgraph F, each component of which has order at least α. We show that if the degree sum of two nonadjacent vertices is greater than ${{2(\vert {G} \vert-\alpha-1)} \over {k+1}}$ then G has a connected subgraph in which F is contained and every vertex $v$ has degree at most ${\rm deg}_{{F}}(v)\,+\,{{k}}$. From the result, we derive that a graph G has a connected $[{{a}},{{b}}]$‐factor $({{b}}>{{a}})$ if the degree sum of two nonadjacent vertices is at least ${{2[(a+1)\vert{G}\vert-(a+2)]} \over {a + b}}$. © Wiley Periodicals, Inc. J. Graph Theory 56: 241–248, 2007

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1≤ d2≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

Bounding the number of cycles in a graph in terms of its degree sequence

Let G be a simple graph of order n. A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in G are co-neighbour vertices if they share the same neighbours. Clearly, if S is a set of pairwise co-neighbour vertices of a graph G, then S is an independent set of G. Let c=a+b√m and c=a−b√m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. In [M. Lepovic, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730-738, 2007], the author defined the matrix Ac(G)=[cij]n to be the conjugate adjacency matrix of G, if cij=c for any two adjacent vertices i and j, cij=c for any two nonadjacent vertices i and j,and cij= 0 if i=j. In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices.

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$n?3 and W a subset of V(G) with $$|W|\ge 3k$$|W|?3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)?2n/3 for each $$x\in W$$x?W, then G contains k vertex-disjoint cycles such that each of them contains at least three vertices of W. In this paper, we obtain an analogue result of Wang's Theorem in bipartite graph with the partial degree condition. Let $$G=(V_1,V_2;E)$$G=(V1,V2?E) be a bipartite graph with $$|V_1|=|V_2|=n$$|V1|=|V2|=n, and let W be a subset of $$V_1$$V1 with $$|W|\ge 2k$$|W|?2k, where k is a positive integer. We show that if $$d(x)+d(y)\ge n+k$$d(x)+d(y)?n+k for every pair of nonadjacent vertices $$x\in W, y\in V_2$$x?W,y?V2, then G contains k vertex-disjoint cycles such that each of them contains at least two vertices of W.

A simple signed graph \((G, \varSigma )\) is a simple graph with a \(+\)ve or a −ve sign assigned to each of its edges where \(\varSigma \) denotes the set of −ve edges. A cycle is unbalanced if it has an odd number of −ve edges. A vertex subset R of \((G, \varSigma )\) is a relative signed clique if each pair of non-adjacent vertices of R is part of an unbalanced 4-cycle. The relative signed clique number \(\omega _{rs}((G, \varSigma ))\) of \((G,\varSigma )\) is the maximum value of |R| where R is a relative signed clique of \((G,\varSigma )\). Given a family \(\mathcal {F}\) of signed graphs, the relative signed clique number is \(\omega _{rs}(\mathcal {F}) = \max \{\omega _{rs}((G,\varSigma ))|(G,\varSigma ) \in \mathcal {F}\}\). For the family \(\mathcal {P}_3\) of signed planar graphs, the problem of finding the value of \(\omega _{rs}(\mathcal {P}_3)\) is an open problem. In this article, we close it by proving \(\omega _{rs}(\mathcal {P}_3)=8\).

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.

Let G be a simple graph and k\in\mathbb{N} . Two important graph operations are: the k -th graph power of G , denoted by G^{k} , where G^{k} is the graph obtained from G by adding an edge between every pair of vertices that have a distance at most k , and the complement graph of G , denoted by \overline{G} . In this paper, we studied the relation between the independence numbers of \overline{G}^{k} and \overline{G^k} .

A new condition on dominated pair degree sum for a digraph to be supereulerian