Let M be a set of k independent edges in a k-connected graph G. If k is odd and M is an edge cut of G, then G clearly does not contain a cycle through all edges of M. Lovasz conjectured that if k is even or G M is connected, then G contains a cycle through all edges of M. In this paper, we present an elegant proof of a weaker version of the Lovasz Conjecture for k-connected claw-free graphs. Specifically, we demonstrate that in a kconnected claw-free graph G, where k is even, if there exists an edge in M not contained in a triangle, then G contains a cycle through all edges of M. This result implies that in a k-connected claw-free graph, there exists a cycle containing an edge cut consisting of any k independent edges, where k is even.
A dominating set in a graph G is a set S of vertices of G such that every vertex in V(G) \ S has a neighbor in S, where two vertices are neighbors if they are adjacent. A total dominating set in G is a dominating set with the additional property that the subgraph G[S] induced by S is isolate-free. For k >= 1 an integer, the k-component domination number gamma(k)(G), first defined in 2016 by Alvarado, Dantas, and Rautenbach, is the minimum cardinality among all dominating sets S of G such that every component in G[S] has order at least k. We note that k-component domination is a natural generalization of both domination and total domination as gamma(1)(G) = gamma(G) and gamma(2)(G) = gamma(t)(G), where gamma(G) is the domination number of G and gamma(t)(G) is the total domination number of G. We observe that for k >= 3 if G is a connected graph of order at least k and gamma(k)(G) > k, then gamma(k)(G) <= 3 gamma(G) - 2 and gamma(k)(G) <= 2 gamma(t)(G) - 2. For k is an element of {3, 4}, we establish properties of connected graphs G of order at least k that satisfy gamma(k)(G) = 3 gamma(G) - 2, and for k is an element of {4, 5, 6}, we establish properties of connected graphs G of order at least k that satisfy gamma(k)(G) = 2 gamma(t)(G) - 2. In addition, for each of these bounds, we characterize the extremal trees.
For two graphs G and H, we call H a {P-2, P-5}-factor of G if H is a spanning subgraph of G with each component isomorphic to P-2 or P-5, where Pi is a path of order i for i = 2, 5. For a positive integer k, a graph G is called ({P-2, P-5}, k)-factor critical if G-V ' contains a {P-2, P-5}-factor for any V ' subset of V(G) with |V '|= k. In this note, we give some sufficient conditions, with respect to binding number and degree sum of non-adjacent vertices, for a graph to contain a {P-2, P-5}-factor or to be ({P-2, P-5}, k)-factor critical, which improve some known results.
This paper introduces new constructions of nonregular cospectral signed graphs via two operations: the neighbours splitting (NS) join and the nonneighbours splitting (NNS) join. We compute the adjacency and Laplacian characteristic polynomials for each join, allowing spectral analysis for arbitrary signed graphs and explicit eigenvalue calculations for co-regular signed graphs. A second approach employs pseudo-potential functions to define more robust switching-stable versions of these joins, preserving spectral properties under switching equivalence. As an application of these techniques, we construct infinite families of nonisomorphic signed graphs exhibiting cospectrality for both adjacency and Laplacian matrices. We also characterise balancedness conditions for each of the constructions.
A two-player edge-selection game played on finite simple graphs is considered, where the objective is to be the first to form a subgraph isomorphic to K-1,K-3. The graphs on which the first player can win in exactly three moves are fully characterized as well as the graphs where she wins if allowed two consecutive initial moves. We also explore graphs with bounded maximum degree, providing constructions and strategies for degrees three and four.
For an integer s >= 0, a graph G of order at least s + 3 is s-Hamiltonian (respectively, s-Hamiltonian-connected) if, for any vertex subset S subset of V (G) with ISI <= s, the graph G-S is Hamiltonian (respectively, Hamiltonian connected). Let G be a graph. In 2019, Lai et al. conjectured that for any s >= 1, the line graph L(G) is (s + 1)-Hamiltonian if and only if kappa(L(G)) >= s + 3 if and only if L(G) is s-Hamiltonian-connected, where kappa(L(G)) is the connectivity of L(G). In this paper, we show that this conjecture holds for the line graph of a planar graph. A graph G is called s-edge-Hamiltonian-connected if for any U subset of {u(1)u(2) : u(1), u(2) E V (G), u(1) =/ u(2)} such that the graph induced by U is a linear forest with 1 <= IUI <= s, the graph G U U contains a Hamiltonian cycle that includes every edge of U, where G U U denotes the graph obtained from G by adding the edges in U. We also show that if G is a planar graph, then L(G) is 2-edge-Hamiltonian-connected if and only if kappa(L(G)) >= 4.
This paper advances the study of connectivity preservation in k-connected graphs by addressing Mader's conjecture on the existence of non-separating trees. We prove that for any k >= 1 and m > 4, every k-connected graph G with minimum degree delta(G) > [(3k) (2)] +m-1 contains a tree T (3) (m-3)-a star-path hybrid structure-such that G-V (T-m-3(3)) remains k-connected. Additionally, we establish the conjecture for the star K-1,K-m-1 and more general trees Ttm-t within a specialized family g of k-connected graphs, where kappa(G(0)(X)) <= k+1 for any subgraph G(0) subset of G and vertex subset X subset of V(G(0)) with |X| = k. Our approach relies on constructing (k + 1)-connected substructures and analyzing their properties to ensure the preservation of k-connectivity after tree removal. These results generalize and strengthen prior work on connectivity keeping trees, providing deeper insights into the structural robustness of highly connected graphs.
A set S subset of V of vertices in a graph G = (V, E) is dominating set of G if every vertex in V \ S has a neighbor in S. If, in addition, every vertex in S also has a neighbor in S, then S is a total dominating set of G. A set S subset of V is a dual-server dominating set if S can be partitioned into two subsets R and B such that every vertex in V \ S is adjacent to at least one vertex in R and at least one vertex in B. In this paper, we introduce and study two distinct definitions for the total version of dual-server domination. Specifically, let S be a dual-server dominating set with a partition {R, B} of S. If every vertex in S has a neighbor in S, then S is called a dual-server total dominating set; while if every vertex in R has a neighbor in R and every vertex in B has a neighbor in B, then S is called a total dual-server dominating set.
A simple connected graph G with vertex set V (G) and edge set E(G) is Z(k)-antimagic if there exists a function f : E(G)-> Z(k)\{0} such that the induced function f(+)(v) = Sigma(uv is an element of E(G)) f(uv) is injective. The integer-antimagic spectrum of a graph G is the set IAM(G) = {k : G is Z(k)-antimagic and k >= 2}. In this paper, we prove that IAM(G) subset of IAM(G '), where G ' is any graph obtained by adding simple edges to G (not equal to a 3-path). Furthermore, if G is disconnected and the added edges do not create a new K-3-component in G ', then IAM(G) C IAM(G ').
A dominating set of a graph G is a set D subset of V (G) such that every vertex in V (G)\D has a neighbor in D, where two vertices are neighbors if they are adjacent. The domination number of G, denoted by gamma(G), is the minimum cardinality among all dominating sets of G. A packing of a graph G is a set of vertices that are mutually distance at least 3 apart. The packing number of G, denoted by rho(G), is the maximum cardinality among all packings of G. It is conjectured that gamma(G) <= 2 rho(G) if G is a connected graph with maximum degree at most 3, except for three graphs. It has also been shown that if G is a claw-free graph with maximum degree at most 3, then gamma(G) <= 2 rho(G). In this paper, we show that gamma(G) <= 2 rho(G) if G is a connected P-5-free graph with maximum degree at most 3. We further show that if G is a connected H-free graph for some H is an element of {P-3 boolean OR P-2, 2K(2) boolean OR K-1} with maximum degree at most 3 except for some finite set of graphs, then gamma(G) <= 2 rho(G).
A connected vertex subset X of a graph G is contractible if G - X is 2-connected. A far-reaching conjecture by McCuaig and Ota states that for every k >= 2 there is a number f(k) such that every 3-connected graph of order at least f(k) contains a contractible set of size k. Here, we study the number of contractible sets of different sizes in 3-connected graphs that are planar and utilize Schnyder woods for this task. It is well-known that Schnyder woods can be used to find three compatible ordered path partitions. We use the latter to prove that every 3and in addition tight. Our results can be seen as support to an affirmative solution of the McCuaig-Ota conjecture for planar graphs.
Let G = (V, E) be a graph and k >= 1. A subset S of V is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. We denote minimum cardinality of a k-path vertex cover in G by psi(k)(G) and call it the k-path vertex cover number of G. A set D subset of V of vertices of G is said to be a distance k-dominating set of G if the distance between each vertex u is an element of V \ D and D is at most k. In the paper we study a relationship between the k-path vertex cover and the k-distance domination number of trees. Moreover, we present a full characterization of trees such that each vertex belongs to some minimum k-path vertex cover.
