Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

On characterizing Vizing's edge colouring bound

We consider multigraphs G for which equality holds in Vizing's classical edge colouring bound χ′(G)≤Δ + µ, where Δ denotes the maximum degree and µ denotes the maximum edge multiplicity of G. We show that if µ is bounded below by a logarithmic function of Δ, then G attains Vizing's bound if and only if there exists an odd subset S?V(G) with |S|≥3, such that |E[S]|>((|S| ? 1)/2)(Δ + µ ? 1). The famous Goldberg–Seymour conjecture states that this should hold for all µ≥2. We also prove a similar result concerning the edge colouring bound χ′(G)≤Δ + ?µ/?g/2??, due to Steffen (here g denotes the girth of the underlying graph). Finally we give a general approximation towards the Goldberg‐Seymour conjecture in terms of Δ and µ. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:160‐168, 2012     

A precolouring extension of Vizing's theorem

Fix a palette of colors, a graph with maximum degree , and a subset of the edge set with minimum distance between edges at least . If the edges of are arbitrarily precoloured from , then there is guaranteed to be a proper edge-coloring using only colors from that extends the precolouring on to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the distance can be lowered to when the graph contains no cycle of length .     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

Improvements on Hippchen's conjecture

Let G be a k-connected graph on n vertices. Hippchen's Conjecture (2008) states that two longest paths in G share at least k vertices. Gutiérrez (2020) recently proved the conjecture when $k\le 4$ or $k\ge \frac{n?2}{3}$. We improve upon both results; namely, we show that two longest paths in G share at least k vertices when $k=5$ or $k\ge \frac{n+2}{5}$. This completely resolves two conjectures by Gutiérrez in the affirmative.     

A survey of selected recent results on total domination in graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. In this paper, we offer a survey of selected recent results on total domination in graphs.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Graham's pebbling conjecture on products of cycles

Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)≤ f(G)f(H). We prove Graham's conjecture when G is a cycle for a variety of graphs H, including all cycles. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 141–154, 2003     

Proof of a conjecture on connectivity of Kronecker product of graphs

For a graph G, κ(G) denotes its connectivity. The Kronecker product G₁×G₂ of graphs G₁ and G₂ is the graph with the vertex set V(G₁)×V(G₂), two vertices (u₁,v₁) and (u₂,v₂) being adjacent in G₁×G₂ if and only if u₁u₂∈E(G₁) and v₁v₂∈E(G₂). Guji and Vumar [R. Guji, E. Vumar, A note on the connectivity of Kronecker products of graphs, Appl. Math. Lett. 22 (2009) 1360–1363] conjectured that for any nontrivial graph G, κ(G×K_n)=min{nκ(G),(n−1)δ(G)} when n≥3. In this note, we confirm this conjecture to be true.     

The ultra‐weak Ash conjecture and some particular cases

Ash's functions N _{σ ,k} count the number of k ‐equivalence classes of σ ‐structures of size n . Some conditions on their asymptotic behavior imply the long standing spectrum conjecture. We present a new condition which is equivalent to this conjecture and we discriminate some easy and difficult particular cases. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

笛卡尔乘积图的成对控制

Let γpr（G） denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr（G）γpr（H）≤ 7γpr （G □H）.     

Light leaves and Lusztig's conjecture

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F  : if the image p=F(l)$p=F(l)$ of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p. We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

On the full Brouwer's Laplacian spectrum conjecture

Let G be a simple connected graph and let $S_k(G)$ be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that $S_k(G)\le e(G)+\left(\begin{array}{c}k+1\\ 2\end{array}\right)$ holds for $1\le k\le n?1$. The case $k=2$ was proved by Haemers, Mohammadian and Tayfeh-Rezaie [Linear Algebra Appl., 2010]. In this paper, we propose the full Brouwer's Laplacian spectrum conjecture and we prove the conjecture holds for $k=2$ which also confirm the conjecture of Guan et al. in 2014.     

Hadwiger's conjecture for quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains K_{t + 1} as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008     

Haglund's conjecture for multi-t Macdonald polynomials

We provide new approaches to prove identities for the modified Macdonald polynomials via their LLT expansions. As an application, we prove a conjecture of Haglund concerning the multi-t-Macdonald polynomials of two rows.