Abstract: Let F be an edge subset and $$F^{\prime }$$ a subset of edges and vertices of a graph G. If $$G-F$$ and $$G-F^{\prime }$$ have no fractional perfect matchings, then F is a fractional matching preclusion (FMP) set and $$F^{\prime }$$ is a fractional strong MP (FSMP) set of G. The FMP (FSMP) number of G is the minimum size of FMP (FSMP) sets of G. In this paper, the FMP number and the FSMP number of Petersen graph, complete graphs and twisted cubes are obtained, respectively.

Abstract: The packing chromatic number $$\chi _{\rho }(G)$$ of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets $$\Pi _1,\ldots ,\Pi _k$$ , where $$\Pi _i$$ , $$i\in [k]$$ , is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then $$\chi _{\rho }(S(G))\le 5$$ , where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if G is a generalized prism of a cycle, then G is (1, 1, 2, 2)-colorable if and only if G is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two n-cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered.

Abstract: A graph is hypohamiltonian if it is not Hamiltonian, but the deletion ofany single vertex gives a Hamiltonian graph. Until now, the smallest knownplanar hypohamiltonian graph had 42 vertices, a result due to Araya andWiener. That result is here improved upon by 25 planar hypohamiltoniangraphs of order 40, which are found through computer-aided generation ofcertain families of planar graphs with girth 4 and a xed number of 4-faces.It is further shown that planar hypohamiltonian graphs exist for all ordersgreater than or equal to 42. If Hamiltonian cycles are replaced by Hamilto-nian paths throughout the de nition of hypohamiltonian graphs, we get thede nition of hypotraceable graphs. It is shown that there is a planar hypo-traceable graph of order 154 and of all orders greater than or equal to 156.We also show that the smallest hypohamiltonian planar graph of girth 5 has45 vertices. Email addresses: mohammadreza@jooyandeh.info (Mohammadreza Jooyandeh),bdm@cs.anu.edu.au (Brendan D. McKay), patric.ostergard@aalto.fi (Patric R. J.Ostergard), ville.pettersson@aalto.fi (Ville H. Pettersson),czamfirescu@gmail.com (Carol T. Zam rescu)URL: http://www.jooyandeh.info (Mohammadreza Jooyandeh),http://cs.anu.edu.au/~bdm (Brendan D. McKay)Preprint submitted to Journal of Combinatorial Theory, Series B February 13, 2013

Abstract: For integers k ≥1 and n≥2k+1 the Kneser graph K(n;k) has as vertices all k-element subsets of [n]:={1;2;:::;n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n,k) has as vertices all k-element and (n—k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph K(5, 2) have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs H(n,k). We also establish the existence of cycles that visit almost all vertices in Kneser graphs K(n,k) when n=2k+o(k), generalizing and improving upon previous results on this problem.

Abstract: Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function of a directed graph, which agrees with the Shareshian-Wachs definition in the acyclic case. We give an F-basis expansion for all digraphs in terms of a permutation statistic, which we call G-descents. We use this expansion to derive a p-positivity formula for all digraphs with symmetric chromatic quasisymmetric functions. We show that the chromatic quasisymmetric functions of a certain class of digraphs, called circular indifference digraphs, have symmetric coefficients. We present an e-positivity formula for the chromatic quasisymmetric function of the directed cycle, which is a t-analog of a result of Stanley. Lastly, we give a generalization of the Shareshian-Wachs e-positivity conjecture to a larger class of digraphs.

Abstract: In this paper, we show that for all except n=2k+2, where is a generalized Petersen graph and is the 2-rainbow domination number of . We also conjecture that if and only if k=1, or n=2k+2 for , or ,...

Abstract: A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$ vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r \rfloor$-weak bisection. In this paper we study problems related to the existence of $k$-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.

Abstract: Graph data management has revealed beneficial characteristics in terms of flexibility and scalability by differently balancing between query expressivity and schema flexibility. This has resulted into an rapid developing new task specific graph systems, query languages and data models, such as property graphs, key-value, wide column, resource description framework (RDF), etc. Present day graph query languages are focused towards flexible graph pattern matching (aka sub-graph matching), where as graph computing frameworks aim towards providing fast parallel (distributed) execution of instructions. The consequence of this rapid growth in the variety of graph based data management systems has resulted in a lack of standardization. Gremlin, a graph traversal language and machine, provides a common platform for supporting any graph computing system (such as an OLTP graph database or OLAP graph processors). We present a formalization of graph pattern matching for Gremlin queries. We also study, discuss and consolidate various existing graph algebra operators into an integrated graph algebra.

Abstract: Results are obtained that substantially strengthen a previously known bound for the chromatic number of a random subgraph of the Kneser graph.

Abstract: Let $G$ be an undirected simple, connected graph with $n \geq 3$ vertices and $m$ edges, with vertex degree sequence $d_1\ge d_2 \ge \cdots \ge d_n$. The general Randi\'c index is defined by \[ R_{-1}=\sum _{(i,j)\in E}\frac {1}{d_id_j}. \] Lower and upper bounds for $R_{-1}$ are obtained in this paper.

Abstract: The first Zagreb index of a graph is defined as the sum of the squares of the degrees of the vertices in the graph. Using first Zagreb index of a graph, we in this note present sufficient conditions for some Hamiltonian properties of the line graph of a graph.

Abstract: A celebrated problem in network optimization is the all-terminal reliability maximization. We want to communicate a fixed number n of terminals, but we have a fixed budget constraint m. The goal is to build m links such that the all-terminal reliability is maximized in the resulting graph. In such case, the result is a uniformly most-reliable graph. The discovery of these graphs is a challenging problem that launched an interplay between extremal graph theory and computational optimization.

Abstract: Given a graph $G$, denote by $\Delta$ and $\chi^\prime$ the maximum degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. We proved that every edge chromatic critical graph of order $n$ with maximum degree at least $\frac{2n}{3}+12$ is Hamiltonian.

Abstract: Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph \(\Gamma _{G}(N)\), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph consists in exactly one triangle.

Abstract: A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.

Abstract: In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If $\sigma_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t \geq 5$, then $G$ contains $k$ disjoint cycles.

Abstract: A graph G is connected if given any two vertices, there is a path between them. A bond B is a minimal edge set in G such that G − B has more components than G. We say that a connected graph is dual Hamiltonian if its largest bond has size |E(G)|−|V (G)|+2. In this thesis we verify the conjecture that any simple 3-connected graph G has a largest bond with size at least Ω(nlog32) (Ding, Dziobiak, Wu, 2015 [3]) for a variety of graph classes including planar graphs, complete graphs, ladders, Mo bius ladders and circular ladders, complete bipartite graphs, some unique (3,g)- cages, the generalized Petersen graph, and some small hypercubes. We will also go further to prove that a variety of these graph classes not only satisfy the conjecture, but are also dual Hamiltonian.

Abstract: A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

Abstract: Hadwiger's conjecture asserts that every graph with chromatic number t contains a complete minor of order t. Given integers n≥2k+1≥5, the Kneser graph K(n,k) is the graph with vertices the k-subsets of an n-set such that two vertices are adjacent if and only if the corresponding k-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.

Abstract: Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topological indices.

Abstract: With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved to be NP-hard and fixed-parameter polynomial algorithms have been obtained for some special classes of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact optimality representations for Petersen graph, the complete k-partite graphs, split graphs, generalized convex graphs, and several planar grids, including rectangular grids, triangular grids, and triangular-rectangular grids.

Abstract: We consider graphs G with Δ=3 such that χ′(G)=4 and χ′(G−e)=3 for every edge e, so-called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, P∗, is such a graph and has average degree only 83. He showed that every critical graph has average degree at least 83, and asked if P∗ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if G is a subcubic critical graph other than P∗, then G has average degree at least 4617≈2.706. This bound is best possible, as shown by the Hajos join of two copies of P∗.

Abstract: Given a commutative ring R with identity 1, its Jacobson graph 𝔍R is defined to be the graph in which the vertex set is R∖J(R), and two distinct vertices x and y are adjacent if and only if 1 − xy∉U(R). Here J(R) denotes the Jacobson radical of R and U(R) is the set of unit elements in R. This paper investigates the genus properties of Jacobson graph. In particular, we determine all isomorphism classes of commutative rings whose Jacobson graph has genus two.

Abstract: A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by Erdős, Pyber and Gyarfas that states that there exists a partition into $r-1$ monochromatic connected subgraphs. We consider a variant of this problem, where the complete graph is replaced by a graph with large minimum degree, and prove two conjectures of Bal and DeBiasio, for two and three colours.

Abstract: We prove that the number of 1‐factorizations of a generalized Petersen graph of the type GP(3k,k) is equal to the kth Jacobsthal number J(k) when k is odd, and equal to 4J(k) when k is even. Moreover, we verify the list coloring conjecture for GP(3k,k) .