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Register a new userThe 1/3-2/3 Conjecture for ordered sets whose cover graph is a forest
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Imed Zaguia
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A balanced pair in an ordered set $P=(V,leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We define the notion of a good pair and claim any ordered set that has a good pair will satisfy the conjecture and furthermore every ordered set which is not totally ordered and has a forest as its cover graph has a good pair.
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The 1-2-3 Conjecture, posed by Karo{n}ski, {L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The 1-2 Conjecture states that if vertices also receive colors and the vertex color is added to the sum of its incident edges, then adjacent vertices can be distinguished using only ${ 1,2}$. In this paper we confirm 1-2 Conjecture for 3-regular graphs. Meanwhile, we show that every 3-regular graph can achieve a neighbor sum distinguishing edge coloring by using 4 colors, which answers 1-2-3 Conjecture positively.
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription. Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lovasz et al.(2013) showed that both conjectures hold if the edge connectivity condition is relaxed to $6$-edge-connected. Both problems are still open for $5$-edge-connected graphs. The $3$-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grotzschs Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al.(2016) provided the first direct proof using flows of the Strong $3$-Flow Conjecture for planar graphs. We prove the Strong $3$-Flow Conjecture for projective planar graphs.
A ${00,01,10,11}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that such a function must satisfy $tle 2n/3-1$; moreover, the $(2n/3-1)$-resilient $(n,2)$-functions correspond to the equitable partitions of the $n$-cube with the quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. We suggest constructions of such functions and corresponding partitions, show connections with Latin hypercubes and binary $1$-perfect codes, characterize the non-full-rank and the reducible functions from the considered class, and discuss the possibility to make a complete characterization of the class.
A conjecture of Graver from 1991 states that the generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic $d$-dimensional rigidity matroid and the generic $C_{d-2}^{d-1}$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_{d-2}^{d-1}$-cofactor matroid is the unique maximal abstract $d$-rigidity matroid for all $dgeq 2$. We verify the case $d=3$ of Whiteleys conjecture in this paper. A key step in our proof is to verify a second conjecture of Whiteley that the `double V-replacement operation preserves independence in the generic $C_2^1$-cofactor matroid.
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